Jordan Algebraic Interpretation of Maximal Parabolic Subalgebras : Exceptional Lie Algebras

TL;DR

This paper explores the connection between Jordan algebras and the structure of maximal parabolic subalgebras in non-compact real forms of exceptional Lie algebras, focusing on Jordan algebras of rank 2 and 3.

Contribution

It introduces a novel Jordan algebraic interpretation of maximal parabolic subalgebras in exceptional Lie algebras, linking two major mathematical areas.

Findings

Jordan algebraic structures correspond to maximal parabolic subalgebras

Focus on non-compact real forms of exceptional Lie algebras

Analysis of Jordan algebras of rank 2 and 3

Abstract

With this paper we start a programme aiming at connecting two vast scientific areas: Jordan algebras and representation theory. Within representation theory, we focus on non-compact, real forms of semisimple Lie algebras and groups as well as on the modern theory of their induced representations, in which a central role is played by the parabolic subalgebras and subgroups. The aim of the present paper and its sequels is to present a Jordan algebraic interpretations of maximal parabolic subalgebras. In this first paper, we confine ourselves to maximal parabolic subalgebras of the non-compact real forms of finite-dimensional exceptional Lie algebras, in particular focusing on Jordan algebras of rank 2 and 3.

Tables

Table 1. Table 1: Lie algebras associated to cubic (i.e., rank-3) Euclidean Jordan algebras. The notation g ​ ( ℂ ) ℝ 𝑔 subscript ℂ ℝ g(\mathbb{C})_{\mathbb{R}} means the algebra g ​ ( ℂ ) 𝑔 ℂ g(\mathbb{C}) seen as a real algebra

Table 2. Table 2: The single-split (non-symmetric) real form of the Magic Square ℒ 3 ​ ( 𝔸 s , 𝔹 ) subscript ℒ 3 subscript 𝔸 𝑠 𝔹 {\mathcal{L}}_{3}(\mathbb{A}_{s},\mathbb{B}) [ 35 ]

	$ℝ$	$ℂ$	$ℍ$	$𝕆$
$ℝ$	$so\left(3\right)$	$su\left(3\right)$	$usp\left(3\right)$	$F_{4\left(-52\right)}$
${ℂ}_s$	$sl(3,ℝ)$	$sl{(3,ℂ)}_{ℝ}$	$s{u}^{\ast }\left(6\right)$	$E_{6\left(-26\right)}$
${ℍ}_s$	$sp(3,ℝ)$	$su(3,3)$	$s{o}^{\ast }\left(12\right)$	$E_{7\left(-25\right)}$
${𝕆}_s$	$F_{4\left(4\right)}$	$E_{6\left(2\right)}$	$E_{7\left(-5\right)}$	$E_{8\left(-24\right)}$

Table 3. Table 3: The double-split (symmetric) real form of the Magic Square ℒ 3 ​ ( 𝔸 s , 𝔹 s ) subscript ℒ 3 subscript 𝔸 𝑠 subscript 𝔹 𝑠 {\mathcal{L}}_{3}(\mathbb{A}_{s},\mathbb{B}_{s}) [ 37 ]

	$ℝ$	${ℂ}_s$	${ℍ}_s$	${𝕆}_s$
$ℝ$	$so\left(3\right)$	$sl(3,ℝ)$	$sp(3,ℝ)$	$F_{4\left(4\right)}$
${ℂ}_s$	$sl(3,ℝ)$	$sl(3,ℝ)\oplus sl(3,ℝ)$	$sl(6,ℝ)$	$E_{6\left(6\right)}$
${ℍ}_s$	$sp(3,ℝ)$	$sl(6,ℝ)$	$so(6,6)$	$E_{7\left(7\right)}$
${𝕆}_s$	$F_{4\left(4\right)}$	$E_{6\left(6\right)}$	$E_{7\left(7\right)}$	$E_{8\left(8\right)}$

Table 4. Table 4: The compact, real form of the Freudenthal-Rozenfeld-Tits symmetric Magic Square ℒ 3 ​ ( 𝔸 , 𝔹 ) subscript ℒ 3 𝔸 𝔹 {\mathcal{L}}_{3}(\mathbb{A},\mathbb{B}) [ 24 ]

	$ℝ$	$ℂ$	$ℍ$	$𝕆$
$ℝ$	$so\left(3\right)$	$su\left(3\right)$	$usp\left(3\right)$	$F_{4\left(-52\right)}$
$ℂ$	$su\left(3\right)$	$su\left(3\right)\oplus su\left(3\right)$	$su\left(6\right)$	$E_{6\left(-78\right)}$
$ℍ$	$usp\left(3\right)$	$su\left(6\right)$	$so\left(12\right)$	$E_{7\left(-133\right)}$
$𝕆$	$F_{4\left(-52\right)}$	$E_{6\left(-78\right)}$	$E_{7\left(-133\right)}$	$E_{8\left(-248\right)}$

Equations

$${\Delta }_{{\mathcal{A}}_0}\doteq \{\lambda \in {\mathcal{A}}_0^{\ast }|\lambda \ne 0,{\mathcal{G}}_{{\mathcal{A}}_0}^{\lambda }\ne 0\},{\mathcal{G}}_{{\mathcal{A}}_0}^{\lambda }\doteq \{X\in \mathcal{G}|[Y,X]=\lambda (Y)X,\forall Y\in {\mathcal{A}}_0\}.$$

$${\mathcal{N}}^{\pm }\doteq \underset{\lambda \in {\Delta }_{{\mathcal{A}}_0}^{\pm }}{\oplus }{\mathcal{G}}_{{\mathcal{A}}_0}^{\lambda }.$$

$$\mathcal{G}={\mathcal{N}}^{+}\oplus {\mathcal{M}}_0\oplus {\mathcal{A}}_0\oplus {\mathcal{N}}^{-},$$

$${\mathcal{P}}^{\prime }={\mathcal{M}}^{\prime }\oplus {\mathcal{A}}^{\prime }\oplus {\mathcal{N}}^{\prime -},$$

$$\mathcal{G}={\mathcal{N}}^{\prime +}\oplus {\mathcal{A}}^{\prime }\oplus {\mathcal{M}}^{\prime }\oplus {\mathcal{N}}^{\prime -},$$

$$\lambda =\sum _{i=1}^r{n}_i{\lambda }_i,{\lambda }_i\in {\Delta }_{{\mathcal{A}}_0}^S,\mathrm{all}{n}_i\ge 0,(resp.\mathrm{all}{n}_i\le 0).$$

$${\mathcal{P}}_{\Theta }={\mathcal{P}}_0\oplus {\mathcal{N}}^{+}(\Theta ),{\mathcal{N}}^{+}(\Theta )\doteq \underset{\lambda \in {\Delta }_{\Theta }^{+}}{\oplus }{\mathcal{G}}_{{\mathcal{A}}_0}^{\lambda }.$$

$${\mathcal{N}}_{\Theta }^{+}\doteq \underset{\lambda \in {\Delta }_{{\mathcal{A}}_0}^{+}-{\Delta }_{\Theta }^{+}}{\oplus }{\mathcal{G}}_{{\mathcal{A}}_0}^{\lambda },{\mathcal{N}}_{\Theta }^{-}\doteq \theta {\mathcal{N}}_{\Theta }^{+}.$$

$$\displaystyle{\cal P}_{\Theta}\$$

$$=$$

$$\displaystyle{\cal G}\$$

$$=$$

$${\Theta }_j^{\max }=S_r\backslash \{j\},1\le j\le r.$$

$$\underset{{\alpha }_1}{\circ }---\underset{{\alpha }_2}{\circ }---\underset{{\alpha }_3}{\circ }\text{raise}11.0pt|\text{raise}22.0pt\circ {\alpha }_6---\underset{{\alpha }_4}{\circ }---\underset{{\alpha }_5}{\circ }$$

$${\mathcal{P}}_i^{6(6)}$$

$$\displaystyle{\cal M}^{6(6)}_{1}\$$

$${\mathcal{M}}_2^{6(6)}$$

$$\displaystyle{\cal M}^{6(6)}_{3}\$$

$$\displaystyle{\cal M}^{6(6)}_{4}\$$

$$\underbrace{\underset{{\alpha }_1}{\circ }---\underbrace{\underset{{\alpha }_2}{\circ }---\underset{{\alpha }_3}{\circ }\text{raise}11.0pt|\text{raise}22.0pt\circ {\alpha }_6---\underset{{\alpha }_4}{\circ }}---\underset{{\alpha }_5}{\circ }}$$

$${\mathcal{P}}_i^{6(2)}$$

$${\mathcal{M}}_1^{6(2)}$$

$$\displaystyle{\cal M}^{6(2)}_{2}\$$

$$\displaystyle{\cal M}^{6(2)}_{3}\$$

$$\displaystyle{\cal M}^{6(2)}_{4}\$$

$$\underbrace{\underset{{\alpha }_1}{\circ }---\underset{{\alpha }_2}{\bullet }---\underset{{\alpha }_3}{\bullet }\text{raise}11.0pt|\text{raise}22.0pt\circ {\alpha }_6---\underset{{\alpha }_4}{\bullet }---\underset{{\alpha }_5}{\circ }}$$

$${\mathcal{P}}_i^{6(-14)}$$

$${\mathcal{M}}_1^{6(-14)}$$

$${\mathcal{M}}_2^{6(-14)}$$

$$\underset{{\alpha }_1}{\circ }---\underset{{\alpha }_2}{\bullet }---\underset{{\alpha }_3}{\bullet }\text{raise}11.0pt|\text{raise}22.0pt\bullet {\alpha }_6---\underset{{\alpha }_4}{\bullet }---\underset{{\alpha }_5}{\circ }$$

$${\mathcal{P}}_i^{6(-26)}$$

$${\mathcal{M}}_1^{6(-26)}$$

$$\underset{{\alpha }_1}{\circ }---\underset{{\alpha }_2}{\circ }---\underset{{\alpha }_3}{\circ }\text{raise}11.0pt|\text{raise}22.0pt\circ {\alpha }_7---\underset{{\alpha }_4}{\circ }---\underset{{\alpha }_5}{\circ }---\underset{{\alpha }_6}{\circ }$$

$${\mathcal{P}}_i^{7(7)}$$

$${\mathcal{M}}_1^{7(7)}$$

$${\mathcal{M}}_2^{7(7)}$$

$${\mathcal{M}}_3^{7(7)}$$

Full text

1. 1 Introduction

2. 2 Parabolic Subalgebras

3. 3 Maximal Parabolic Subalgebras of the Exceptional Lie Algebras

4. 3.1 EI :  $E_{6(6)}$

5. 3.2 EII :  $E_{6(2)}$

6. 3.3 EIII :  $E_{6(-14)}$

7. 3.4 EIV :  $E_{6(-26)}$

8. 3.5 EV :  $E_{7(7)}$

9. 3.6 EVI :  $E_{7(-5)}$

10. 3.7 EVII :  $E_{7(-25)}$

11. 3.8 EVIII :  $E_{8(8)}$

12. 3.9 EIX :  $E_{8(-24)}$

13. 3.10 FI :  $F_{4(4)}$

14. 3.11 FII :  $F_{4(-20)}$

15. 3.12 G :  $G_{2(2)}$

16. 3.13 Parabolically related non-compact semisimple Lie algebras

17. 4 Jordan Algebras, Freudenthal Triple Systems, Their Symmetries and Embeddings

18. 4.1 Jordan Algebras

19. 4.2 Jordan Pairs

20. 4.3 Symmetries of Jordan Algebras

21. 4.4 Freudenthal Triple Systems

22. 4.5 Extended Freudenthal Triple Systems

23. 4.6 Embeddings

24. 5 Jordan Structures in Maximal Parabolics of Exceptional Lie Algebras : Analysis

25. 5.1 $E_{6(6)}$

26. 5.1.1 $\mathcal{P}_{1}^{6(6)}\cong\mathcal{P}_{5}^{6(6)}$

27. 5.1.2 ${\cal P}_{2}^{6(6)}$

28. 5.1.3 ${\cal P}_{3}^{6(6)}\cong{\cal P}_{6}^{6(6)}$

29. 5.1.4 ${\cal P}_{4}^{6(6)}$

30. 5.2 $E_{6(2)}$

31. 5.2.1 ${\cal P}_{1}^{6(2)}$

32. 5.2.2 ${\cal P}_{2}^{6(2)}$

33. 5.2.3 ${\cal P}_{3}^{6(2)}$

34. 5.2.4 ${\cal P}_{4}^{6(2)}$

35. 5.3 $E_{6(-14)}$

36. 5.3.1 ${\cal P}_{1}^{6(-14)}$

37. 5.3.2 ${\cal P}_{2}^{6(-14)}$

38. 5.4 $E_{6(-26)}$

39. 5.5 $E_{7(7)}$

40. 5.5.1 ${\cal P}_{1}^{7(7)}$

41. 5.5.2 ${\cal P}_{2}^{7(7)}$

42. 5.5.3 ${\cal P}_{3}^{7(7)}$

43. 5.5.4 ${\cal P}_{4}^{7(7)}$

44. 5.5.5 ${\cal P}_{5}^{7(7)}$

45. 5.5.6 ${\cal P}_{6}^{7(7)}$

46. 5.5.7 ${\cal P}_{7}^{7(7)}$

47. 5.6 $E_{7(-5)}$

48. 5.6.1 ${\cal P}_{1}^{7(-5)}$

49. 5.6.2 ${\cal P}_{2}^{7(-5)}$

50. 5.6.3 ${\cal P}_{3}^{7(-5)}$

51. 5.6.4 ${\cal P}_{4}^{7(-5)}$

52. 5.7 $E_{7(-25)}$

53. 5.7.1 ${\cal P}_{1}^{7(-25)}$

54. 5.7.2 ${\cal P}_{2}^{7(-25)}$

55. 5.7.3 ${\cal P}_{3}^{7(-25)}$

56. 5.8 $E_{8(8)}$

57. 5.8.1 ${\cal P}_{1}^{8(8)}$

58. 5.8.2 ${\cal P}_{2}^{8(8)}$

59. 5.8.3 ${\cal P}_{3}^{8(8)}$

60. 5.8.4 ${\cal P}_{4}^{8(8)}$

61. 5.8.5 ${\cal P}_{5}^{8(8)}$

62. 5.8.6 ${\cal P}_{6}^{8(8)}$

63. 5.8.7 ${\cal P}_{7}^{8(8)}$

64. 5.8.8 ${\cal P}_{8}^{8(8)}$

65. 5.9 $E_{8(-24)}$

66. 5.9.1 ${\cal P}_{1}^{8(-24)}$

67. 5.9.2 ${\cal P}_{2}^{8(-24)}$

68. 5.9.3 ${\cal P}_{3}^{8(-24)}$

69. 5.9.4 ${\cal P}_{4}^{8(-24)}$

70. 5.10 $F_{4(4)}$

71. 5.10.1 ${\cal P}_{1}^{4(4)}$

72. 5.10.2 ${\cal P}_{2}^{4(4)}$

73. 5.10.3 ${\cal P}_{3}^{4(4)}$

74. 5.10.4 ${\cal P}_{4}^{4(4)}$

75. 5.11 $F_{4(-20)}$

76. 5.12 $G_{2(2)}$

77. 6 Outlook

1 Introduction

The aim of this paper is to relate two vast scientific areas: Jordan algebras and representation theory. Jordan algebras are beautiful mathematical structures that arose together with the advent of quantum mechanics [1]–[27]. They are widely used in theoretical and mathematical physics, in particular in supergravity and superstring theory; cf., e.g., [28]–[56]; we will review their role in such frameworks in Sec. 4. Intriguingly, recent developments related the standard model of particle physics with the Albert algebra, namely the exceptional rank-3 simple Jordan algebra over the octonions [57]–[61].

Within the area of representation theory, we focus on non-compact semisimple Lie algebras and groups starting from the treatment of Gelfand [62],[63] and Harish-Chandra [64],[65] (see also [66]), up to the most advanced approach of Langlands [67], later refined in [68]; see also [69],[70]. The main building block of the modern theory of induced representations of non-compact semisimple Lie algebras and groups are the parabolic subalgebras and subgroups, cf., e.g., [71]–[76].

We aim at presenting Jordan algebraic interpretations of maximal parabolic subalgebras of non-compact real forms of semisimple Lie algebras, focussing on rank-3 and rank-2 (simple and semisimple) Jordan algebras. This is a rather lengthy and far reaching project, whose proper treatment deserves to be developed in a series of papers. In the present paper, which inits the series, we confine ourselves to maximal parabolic subalgebras of the non-compact real forms of finite-dimensional exceptional Lie algebras, building over the results and treatment of [77, 78, 79].

The paper is organized as follows.

In Section 2 we present a concise treatment of the general theory of parabolic subalgebras [77, 79]. Then, Section 3 recalls the classification of maximal parabolic subalgebras of all non-compact real forms of finite-dimensional exceptional simple Lie algebras; moreover, in Sec. 3.13 we reconsider the definition of parabolically related non-compact semisimple Lie algebras, and in Table A we present the classification of maximally parabolically related exceptional Lie algebras, slightly extending the results of [78]. An overview of the general theory of rank-3 and rank-2 Jordan algebras and their symmetries and related structures, along with a résumé of their relevance for Maxwell-Einstein (super)gravity theories in various space-time dimensions, is provided in Sec. 4. Such Section also contains Table B, in which the Jordan algebraic interpretation of Table A is provided. The content of Table B is derived in a detailed way in the long Sec. 5, in which the maximal parabolic subalgebras of non-compact real forms of exceptional Lie algebras are analyzed and obtained through sequences of maximal embeddings of Lie algebras, which in turn allow for a natural interpretation in terms of symmetries of Jordan algebras. Finally, a brief summary and outlook to future developments is given in Sec. 6.

2 Parabolic Subalgebras

This section follows [77]. Let $G$ be a noncompact semisimple Lie group. Let ${\cal G}$ be the Lie algebra of $G$, $\theta$ be a Cartan involution in ${\cal G}$, and ${\cal G}={\cal K}\oplus{\cal P}$ be a Cartan decomposition of ${\cal G}$, so that $\theta X=X,~{}X\in{\cal K}$, $\theta X=-X,~{}X\in{\cal P}$ ; ${\cal K}$ is a maximal compact subalgebra of ${\cal G}$; in general, ${\cal P}$ fits in a (reducible) representation of the algebra ${\cal K}$.

Let ${\cal A}_{0}$ be a maximal subspace of ${\cal P}$ which is an abelian subalgebra of ${\cal G}$ ; r=\dim ${\cal A}_{0}$ is the  split  (or real) rank of ${\cal G}$, $1\leq r\leq\ell={\rm rank}\,{\cal G}$. The subalgebra  ${\cal A}_{0}$  is called a Cartan subspace of  ${\cal P}$.

Let ${\Delta}_{{\cal A}_{0}}$ be the root system of the pair $({\cal G},{\cal A}_{0})$:

[TABLE]

The elements of ${\Delta}_{{\cal A}_{0}}$ are called ${\cal A}_{0}\,$-restricted roots. For $\lambda\in{\Delta}_{{\cal A}_{0}}\,$,  ${\cal G}_{{\cal A}_{0}}^{\lambda}$ are called ${\cal A}_{0}\,$-* restricted root spaces*,  dim${}_{R}~{}{\cal G}_{{\cal A}_{0}}^{\lambda}\geq 1$. In a standard way, the ${\cal A}_{0}\,$-* restricted roots* are split into positive and negative restricted roots:  ${\Delta}_{{\cal A}_{0}}={\Delta}_{{\cal A}_{0}}^{+}\cup{\Delta}_{{\cal A}_{0}}^{-}\,$. Then we introduce the corresponding nilpotent subalgebras:

[TABLE]

Next let ${\cal M}_{0}$ be the centralizer of ${\cal A}_{0}$ in ${\cal K}$, i.e., ${\cal M}_{0}\ \doteq\ \{X\in{\cal K}~{}|~{}[X,Y]=0,~{}\forall Y\in{\cal A}_{0}\}$. In general ${\cal M}_{0}$ is a compact reductive Lie algebra.

For the Bruhat decomposition, it holds that [66]:

[TABLE]

and the subalgebra  ${\cal P}_{0}\doteq{\cal M}_{0}\oplus{\cal A}_{0}\oplus{\cal N}^{-}$  called a  minimal parabolic subalgebra  of ${\cal G}$. (Note that we may take equivalently ${\cal N}^{+}$ instead of ${\cal N}^{-}\,$.)

A * standard parabolic subalgebra*  is any subalgebra  ${\cal P}^{\prime}$  of ${\cal G}$ containing  ${\cal P}_{0}\,$. The number of standard parabolic subalgebras, including ${\cal P}_{0}$ and ${\cal G}$, is $2^{r}$.

Thus, if $r=1$ the only nontrivial parabolic subalgebra is ${\cal P}_{0}\,$.

Thus, further in this section $r>1$.

Any standard parabolic subalgebra is of the form:

[TABLE]

so that ${\cal M}^{\prime}\supseteq{\cal M}_{0}$, ${\cal A}^{\prime}\subseteq{\cal A}_{0}$, ${\cal N}^{\prime-}\subseteq{\cal N}^{-}$ ; ${\cal M}^{\prime}$ is the centralizer of ${\cal A}^{\prime}$ in ${\cal G}$ (mod ${\cal A}^{\prime})$ ; ${\cal N}^{\prime-}$  is comprised from the negative root spaces of the restricted root system ${\Delta}_{{\cal A}^{\prime}}$ of $({\cal G},{\cal A}^{\prime})$. The decomposition (2.4) is called the Langlands decomposition of  ${\cal P}^{\prime}\,$. One also has the Bruhat decomposition (2.3) for this general situation:

[TABLE]

where ${\cal N}^{\prime+}=\theta{\cal N}^{\prime-}$.

The standard parabolic subalgebras may be described explicitly using the restricted simple root system  ${\Delta}^{S}_{{\cal A}_{0}}\,$, such that if $\lambda\in{\Delta}_{{\cal A}_{0}}^{+}$, (resp. $\lambda\in{\Delta}_{{\cal A}_{0}}^{-}$), one has:

[TABLE]

We shall follow Warner [69], where one may find all references to the original mathematical work on parabolic subalgebras. For a short formulation one may say that the parabolic subalgebras correspond to the various subsets of ${\Delta}^{S}_{{\cal A}_{0}}$ - hence their number $2^{r}$. To formalize this let us denote:  $S_{r}=\{1,2,\ldots,r\}$, and let  $\Theta$  denote any subset of  $S_{r}\,$. Let  ${\Delta}^{\pm}_{\Theta}\in{\Delta}_{{\cal A}_{0}}$  denote all positive/negative restricted roots which are linear combinations of the simple restricted roots $\lambda_{i}\,,\forall\,i\in\Theta$. Then a standard parabolic subalgebra corresponding to  $\Theta$  will be denoted by  ${\cal P}_{\Theta}$  and is given explicitly as:

[TABLE]

Clearly,  ${\cal P}_{\emptyset}~{}=~{}{\cal P}_{0}\,$,   ${\cal P}_{S_{r}}~{}=~{}{\cal G}\,$,  since  ${\cal N}^{+}(\emptyset)~{}=~{}0\,$,   ${\cal N}^{+}(S_{r})~{}=~{}{\cal N}^{+}\,$. Further, we need to bring (2.7) in the form (2.4). First, define  ${\cal G}(\Theta)$  as the algebra generated by ${\cal N}^{+}(\Theta)$ and ${\cal N}^{-}(\Theta)\doteq\theta{\cal N}^{+}(\Theta)\,$. Next, define ${\cal A}(\Theta)\doteq{\cal G}(\Theta)\cap{\cal A}_{0}$, and  ${\cal A}_{\Theta}$  as the orthogonal complement (relative to the Euclidean structure of ${\cal A}_{0}$) of ${\cal A}(\Theta)$ in ${\cal A}_{0}$. Then  ${\cal A}_{0}={\cal A}(\Theta)\oplus{\cal A}_{\Theta}\,$. Note that  ${\rm dim}\,{\cal A}(\Theta)=|\Theta|$,  ${\rm dim}\,{\cal A}_{\Theta}=r-|\Theta|$. Next, define:

[TABLE]

Then  ${\cal N}^{\pm}={\cal N}^{\pm}(\Theta)\oplus{\cal N}^{\pm}_{\Theta}\,$. Next, define  ${\cal M}_{\Theta}\doteq{\cal M}_{0}\oplus{\cal A}(\Theta)\oplus{\cal N}^{+}(\Theta)\oplus{\cal N}^{-}(\Theta)$. Then ${\cal M}_{\Theta}$ is the centralizer of ${\cal A}_{\Theta}$ in ${\cal G}$ (mod ${\cal A}_{\Theta}$). Finally, we can derive:

[TABLE]

Thus, we have rewritten explicitly the standard parabolic ${\cal P}_{\Theta}$ in the desired form (2.4). The associated (generalized) Bruhat decomposition (2.5) is given now explicitly as:

[TABLE]

In this paper we concentrate on the *maximal * parabolic subalgebras which correspond to $\Theta$ of the form:

[TABLE]

${\rm dim}\,{\cal A}(\Theta^{\rm max}_{j})=r-1$,  ${\rm dim}\,{\cal A}_{\Theta^{\rm max}_{j}}=1$.

3 Maximal Parabolic Subalgebras of the Exceptional Lie Algebras

The material of this Section is taken from [77]. Here we list the real forms of the exceptional simple Lie algebras and their maximal parabolic subalgebras.

First we mention a class of real forms, the maximally non-compact, or  split, real forms, which exist for every complex simple algebra. For the split real forms we can use the same basis as for the corresponding complex simple Lie algebra ${\cal G}^{\mathbb{C}}$, but over $\mathbb{R}$. Restricting $\mathbb{C}\longrightarrow\mathbb{R}$ one obtains the Bruhat decomposition of ${\cal G}$ (with ${\cal M}_{0}=0$) from the triangular decomposition of ${\cal G}^{\mathbb{C}}={\cal G}^{+}\oplus{\cal H}^{\mathbb{C}}\oplus{\cal G}^{-}$, and obtains the minimal parabolic subalgebras ${\cal P}_{0}$ from the Borel subalgebra  ${\cal B}={\cal H}^{\mathbb{C}}\oplus{\cal G}^{+}$, (or ${\cal G}^{-}$ instead of ${\cal G}^{+}$). Furthermore, in this case  ${\rm dim}_{\mathbb{R}}\,{\cal K}~{}=~{}{\rm dim}_{\mathbb{R}}\,{\cal N}^{\pm}$.

For the real forms in general we need to use the Satake diagrams [71, 69]. A Satake diagram has as a starting point the Dynkin diagram of the corresponding complex form. For a split real form it remains the same [69]. In the other cases some dots of the Dynkin diagram are painted in black - these considered by themselves are Dynkin diagrams of the compact semisimple factors  ${\cal M}_{0}$  of the minimal parabolic subalgebras. Further, there are arrows connecting some nodes which use the $\mathbb{Z}_{2}$ symmetry of some Dynkin diagrams. Then the reduced root systems are described by Dynkin-Satake diagrams which are obtained from the Satake diagrams by dropping the black nodes, identifying the arrow-related nodes, and adjoining all nodes in a connected Dynkin-like diagram, but in addition noting the multiplicity of the reduced roots (which is in general different from 1).

Below all dimensions of vector spaces are real.

Note that from now on we shall omit the superscript  max  in order to simplify the notation. Instead we include specification for each maximal parabolics.

3.1 EI :  $E_{6(6)}$

The split real form of  $E_{6}$  is denoted as  $E_{6(6)}$  or  $E^{\prime}_{6}\,$. The maximal compact subgroup is  ${\cal K}\cong sp(4)$, ${\cal P}=\wedge_{0}^{4}\mathbf{8}$,  ${\rm dim}\,{\cal P}=42$,  ${\rm dim}\,{\cal N}^{\pm}={\rm dim}\,{\cal K}=36$. This real form does not have discrete series representations.

Since  $E_{6(6)}$  is split the Satake diagram coincides with the Dynkin diagram:

[TABLE]

Taking into account the above enumeration of simple roots and (2.15) the maximal parabolic subalgebras are determined by [77]:

[TABLE]

There are essentially four maximal cases (instead of six) due to the symmetry of the Satake diagram. The two isomorphic occurrences are distinguished by the fact which node we delete to obtain a maximal parabolic.

Clearly, no maximal parabolic subalgebra is cuspidal.

3.2 EII :  $E_{6(2)}$

Another real form of  $E_{6}$  is denoted as  $E_{6(2)}\,$ or  $E^{\prime\prime}_{6}\,$. The maximal compact subgroup is  ${\cal K}\cong su(6)\oplus su(2)$, ${\cal P}=\left(\mathbf{20},\mathbf{2}\right)$, where $\mathbf{20}=\wedge^{3}\mathbf{6}$,  ${\rm dim}\,{\cal P}=40$,  ${\rm dim}\,{\cal N}^{\pm}=36$,  ${\cal M}_{0}\cong u(1)\oplus u(1)$. This real form has discrete series representations.

The Satake diagram is:

[TABLE]

The split rank is four and thus there are four maximal parabolic subalgebras given by [77]:

[TABLE]

Only the last case in (3.7) is cuspidal, and it has highest/lowest weight representations.

3.3 EIII :  $E_{6(-14)}$

Another real form of  $E_{6}$  is denoted as  $E_{6(-14)}\,$ or  $E^{\prime\prime\prime}_{6}\,$. The maximal compact subgroup is  ${\cal K}\cong so(10)\oplus so(2)$,  ${\cal P}=\mathbf{16}_{-3}\oplus\overline{\mathbf{16}}_{3}$, ${\rm dim}\,{\cal P}=32$,  ${\rm dim}\,{\cal N}^{\pm}=30$,  ${\cal M}_{0}\cong so(6)\oplus so(2)$. This real form has discrete series representations (and highest/lowest weight representations).

The Satake diagram is:

[TABLE]

The split rank is two and thus there are two maximal parabolic subalgebras given by [77]:

[TABLE]

The 2nd is cuspidal and it has highest/lowest weight representations.

3.4 EIV :  $E_{6(-26)}$

Another real form of  $E_{6}$  is denoted as  $E_{6(-26)}\,$ or  $E^{iv}_{6}\,$. This the minimally non-compact real form of $E_{6}$. The maximal compact subgroup is  ${\cal K}\cong f_{4}$, ${\cal P}=\mathbf{26}$,  ${\rm dim}\,{\cal P}=26$,  ${\rm dim}\,{\cal N}^{\pm}=24$,  ${\cal M}\cong so(8)$. This real form does not have discrete series representations.

The Satake diagram is:

[TABLE]

The split rank is equal to 2, thus we have [77]:

[TABLE]

We distinguish the two isomorphic maximal parabolic subalgebras by the fact which noncompact node on the Satake diagram we delete - the first - for  ${\cal M}^{6(-26)}_{1}\,$, or the last, for  ${\cal M}^{6(-26)}_{2}\,$. This case is not cuspidal.

3.5 EV :  $E_{7(7)}$

The split real form of  $E_{7}$  is denoted as  $E_{7(7)}\,$ or  $E^{\prime}_{7}\,$. The maximal compact subgroup is  ${\cal K}\cong su(8)$, ${\cal P}=\wedge^{4}\mathbf{8}$,  ${\rm dim}\,{\cal P}=70$,  ${\rm dim}\,{\cal N}^{\pm}={\rm dim}\,{\cal K}=63$. This real form has discrete series representations.

The Dynkin-Satake diagram is:

[TABLE]

Taking into account the above enumeration of simple roots and (2.15) the maximal parabolic subalgebras are determined by (only the first case is cuspidal) :

[TABLE]

3.6 EVI :  $E_{7(-5)}$

Another real form of  $E_{7}$  is denoted as  $E_{7(-5)}\,$ or $E^{\prime\prime}_{7}\,$. The maximal compact subgroup is  ${\cal K}\cong so(12)\oplus su(2)$, ${\cal P}=\left(\mathbf{32},\mathbf{2}\right)$, where $\mathbf{32}$ is the semispinor irrepr. of $so(12)$,  ${\rm dim}\,{\cal P}=64$,  ${\rm dim}\,{\cal N}^{\pm}=60$,  ${\cal M}\cong su(2)\oplus su(2)\oplus su(2)$. This real form has discrete series representations.

The Satake diagram is:

[TABLE]

The split rank is equal to 4, thus, there are four maximal parabolic subalgebras given by [77]:

[TABLE]

the first case being cuspidal and having highest/lowest weight representations.

3.7 EVII :  $E_{7(-25)}$

Another real form, the minimally non-compact one, of  $E_{7}$  is denoted as  $E_{7(-25)}\,$ or  $E^{\prime\prime\prime}_{7}\,$. The maximal compact subgroup is  ${\cal K}\cong e_{6}\oplus so(2)$, ${\cal P}=\mathbf{27}_{2}\oplus\overline{\mathbf{27}}_{-2}$,  ${\rm dim}\,{\cal P}=54$,  ${\rm dim}\,{\cal N}^{\pm}=51$,  ${\cal M}\cong so(8)$. This real form has discrete series representations (and highest/lowest weight representations).

The Satake diagram is:

[TABLE]

The split rank is equal to 3, thus, there are three maximal parabolic subalgebras given by [77]:

[TABLE]

the first case being cuspidal and having highest/lowest weight representations.

3.8 EVIII :  $E_{8(8)}$

The split real form of  $E_{8}$  is denoted as  $E_{8(8)}\,$ or   $E^{\prime}_{8}\,$. The maximal compact subgroup  ${\cal K}\cong so(16)$, ${\cal P}=\mathbf{128}$ is the semispinor irrepr. of $so(16)$,  ${\rm dim}\,{\cal P}=128$,  ${\rm dim}\,{\cal N}^{\pm}={\rm dim}\,{\cal K}=120$. This real form has discrete series representations.

The Dynkin-Satake diagram is:

[TABLE]

Taking into account the above enumeration of simple roots and (2.15) the maximal parabolic subalgebras are determined by:

[TABLE]

Only the seventh case is cuspidal.

3.9 EIX :  $E_{8(-24)}$

Another real form of  $E_{8}$  is denoted as  $E_{8(-24)}\,$ or  $E^{\prime\prime}_{8}\,$. The maximal compact subgroup is  ${\cal K}\cong e_{7}\oplus su(2)$, ${\cal P}=\left(\mathbf{56},\mathbf{2}\right)$,  ${\rm dim}\,{\cal P}=112$,  ${\rm dim}\,{\cal N}^{\pm}=108$,  ${\cal M}\cong so(8)$. This real form has discrete series representations.

The Satake diagram

[TABLE]

The split rank is equal to 4, thus, there are four maximal parabolic subalgebras given by [77]:

[TABLE]

the last case being cuspidal and having highest/lowest weight representations.

3.10 FI :  $F_{4(4)}$

The split real form of  $F_{4}$  is denoted as  $F_{4(4)}\,$ or  $F^{\prime}_{4}\,$. The maximal compact subgroup  ${\cal K}\cong sp(3)\oplus su(2)$, ${\cal P}=\left(\mathbf{14^{\prime}},\mathbf{2}\right)$, where $\mathbf{14^{\prime}}=\wedge_{0}^{3}\mathbf{6}$,  ${\rm dim}\,{\cal P}=28$,  ${\rm dim}\,{\cal N}^{\pm}={\rm dim}\,{\cal K}=24$. This real form has discrete series representations.

The Dynkin-Satake diagram is:

[TABLE]

The maximal parabolic subalgebras are given by [77] (being $F_{4}$ non-simply laced, we denote the short/long nature of the roots) :

[TABLE]

Note that the last two cases of (3.48) are cuspidal and have highest/lowest weight representations.

Not that  ${\cal P}^{4(4)}_{1}$  and  ${\cal P}^{4(4)}_{2}$  seem isomorphic, however, they are distinguished by the fact what roots generate the  $sl(3,\mathbb{R})$ and $sl(2,\mathbb{R})$ subalgebras of  ${\cal M}^{4(4)}_{1,2}$  - long or short. Also the algebras  ${\cal N}^{4(4)\pm}_{1}$  and  ${\cal N}^{4(4)\pm}_{2}$  differ by the number of long and short roots, as indicated.

3.11 FII :  $F_{4(-20)}$

The other real form of  $F_{4}$  is denoted as  $F_{4(-20)}\,$ or  $F^{\prime\prime}_{4}\,$. The maximal compact subgroup  ${\cal K}\cong so(9)$, ${\cal P}=\mathbf{16}$,  ${\rm dim}\,{\cal P}=16$,  ${\rm dim}\,{\cal N}^{\pm}=15$,  ${\cal M}_{0}~{}\cong~{}so(7)$. This real form has discrete series representations.

The Satake diagram is:

[TABLE]

The split rank is equal to 1, thus, the minimal and maximal parabolics coincide, the  ${\cal M}$-factor and  ${\cal N}$-factor are the same as in the Bruhat decomposition:

[TABLE]

3.12 G :  $G_{2(2)}$

The unique non-compact real form of  $G_{2}$  is the split form, denoted as  $G_{2(2)}$ or  $G^{\prime}_{2}\,$. The maximal compact subgroup  ${\cal K}\cong su(2)\oplus su(2)$, ${\cal P}=\left(\mathbf{4},\mathbf{2}\right)$, where $\mathbf{4}=S^{3}\mathbf{2}$,  ${\rm dim}\,{\cal P}=8$,  ${\rm dim}\,{\cal N}^{\pm}={\rm dim}\,{\cal K}=6$. This real form has discrete series representations.

The Dynkin-Satake diagram is:

[TABLE]

The maximal parabolic subalgebras are given by (being $G_{2}$ non-simply laced, we denote the short/long nature of the roots):

[TABLE]

They are cuspidal and have highest/lowest weight representations.

Not that  ${\cal P}^{2(2)}_{L}$  and  ${\cal P}^{2(2)}_{S}$  seem isomorphic, however, they are distinguished by the fact what root generate the  ${\cal M}^{2(2)}=sl(2,\mathbb{R})$ subalgebras – long or short. Also the algebras  ${\cal N}^{2(2)\pm}_{L}$  and  ${\cal N}^{2(2)\pm}_{S}$  differ by the number of long and short roots, as indicated.

3.13 Parabolically related non-compact semisimple Lie algebras

Next, let us introduce the notion of ‘parabolically related non-compact semisimple Lie algebras’ [78] which is also very useful in the study of the structure of real forms.

Definition:    Let  ${\cal G},{\cal G}^{\prime}$  be two non-compact semisimple Lie algebras with the same complexification  ${\cal G}^{\mathbb{C}}\cong{\cal G}^{\prime\mathbb{C}}$. We call them  parabolically related  if they have parabolic subalgebras  ${\cal P}={\cal M}\oplus{\cal A}\oplus{\cal N}$,  ${\cal P}^{\prime}={\cal M}^{\prime}\oplus{\cal A}^{\prime}\oplus{\cal N}^{\prime}$, such that:  ${\cal M}^{\mathbb{C}}~{}\cong~{}{\cal M}^{\prime\mathbb{C}}$  ($\Rightarrow{\cal P}^{\mathbb{C}}~{}\cong~{}{\cal P}^{\prime\mathbb{C}}$).

Certainly, there may be more than one parabolic relationship for an algebra  ${\cal G}$. Furthermore, two algebras  ${\cal G},{\cal G}^{\prime}$  may be parabolically related with different parabolic subalgebras. In the case the parabolic subalgebras determining the relation are maximal, the two corresponding algebras are said to be maximally parabolically related.

We summarize the maximally parabolically related exceptional Lie algebras in the following table, which slightly extends the results of [78] :

Table A : maximally parabolically related non-compact real forms of finite-dimensional exceptional Lie algebras; the corresponding interpretation in terms of symmetries of rank-2 and rank-3 Jordan algebras is given in Table B

${\cal G}$

${\cal M}$

${\cal G}^{\prime}$

${\cal M}^{\prime}$

${\rm dim}{\cal N}$

1 : $E_{6(-14)}$

$su(5,1)$

$E_{6(6)}$

$sl(6,\mathbb{R})$

$21$

$E_{6(2)}$

$su(3,3)$

2 : $E_{6(-14)}$

$so(7,1)\oplus so(2)$

$E_{6(2)}$

$so(5,3)\oplus u(1)$

$24$

3 : $E_{6(-26)}$

$so(9,1)$

$E_{6(6)}$

$so(5,5)$

$16$

4 : $E_{6(6)}$

$sl(3,\mathbb{R})\oplus sl(3,\mathbb{R})\oplus sl(2,\mathbb{R})$

$E_{6(2)}$

$sl(3,\mathbb{C})_{\mathbb{R}}\oplus sl(2,\mathbb{R})$

$29$

5 : $E_{7(-25)}$

$E_{6(-26)}$

$E_{7(7)}$

$E_{6(6)}$

$27$

6 : $E_{7(-25)}$

$so(9,1)\oplus sl(2,\mathbb{R})$

$E_{7(7)}$

$so(5,5)\oplus sl(2,\mathbb{R})$

$42$

$E_{7(-5)}$

$so(7,3)\oplus su(2)$

7 : $E_{7(-25)}$

$so(10,2)$

$E_{7(7)}$

$so(6,6)$

$33$

$E_{7(-5)}$

$so^{*}(12)$

8 : $E_{7(7)}$

$sl(4,\mathbb{R})\oplus sl(2,\mathbb{R})\oplus sl(3,\mathbb{R})$

$E_{7(-5)}$

$su^{*}(4)\oplus su(2)\oplus sl(3,\mathbb{R})$

$53$

9 : $E_{7(-5)}$

$su^{*}(6)\oplus sl(2,\mathbb{R})$

$E_{7(7)}$

$sl(6,\mathbb{R})\oplus sl(2,\mathbb{R})$

$47$

$E_{7(-25)}$

$su^{*}(6)\oplus su(2)$

10 : $E_{8(-24)}$

$E_{7(-25)}$

$E_{8(8)}$

$E_{7(7)}$

$57$

11 : $E_{8(-24)}$

$so(11,3)$

$E_{8(8)}$

$so(7,7)$

$78$

12 : $E_{8(-24)}$

$E_{6(-26)}\oplus sl(2,\mathbb{R})$

$E_{8(8)}$

$E_{6(6)}\oplus sl(2,\mathbb{R})$

$83$

13 : $E_{8(-24)}$

$so(9,1)\oplus sl(3,\mathbb{R})$

$E_{8(8)}$

$so(5,5)\oplus sl(3,\mathbb{R})$

$97$

14 : $F_{4(-20)}$

$so(7)$

$F_{4(4)}$

$so(4,3)$

$15$

4 Jordan Algebras, Freudenthal Triple Systems, Their Symmetries and

Embeddings

4.1 Jordan Algebras

A Jordan algebra $\mathfrak{J}$ [1, 2, 3, 4, 5] is vector space defined over a ground field $\mathbb{F}$ equipped with a bilinear product (named Jordan product) satisfying

[TABLE]

For the treatment given in the present investigation, the relevant Jordan algebras are examples of the class of cubic Jordan algebras over $\mathbb{F}=\mathbb{R}$ [6, 7, 8, 9]. A cubic Jordan algebra is endowed with a cubic form $N:\mathfrak{J}\rightarrow\mathbb{R}$, such that $N(\lambda X)=\lambda^{3}N(X),\quad\forall\ \lambda\in\mathbb{R},\ X\in\mathfrak{J}$. Moreover, an element $c\in\mathfrak{J}$ exists, satisfying $N(c)=1$ (usually named base point). A general procedure for constructing cubic Jordan algebras, due to Freudenthal, Springer and Tits [10, 11, 12], exists, in which all properties of the Jordan algebra are determined by the cubic form itself.

Let $V$ be a vector space equipped with a cubic norm $N:V\rightarrow\mathbb{R}$ such that $N(\lambda X)=\lambda^{3}N(X),\ \forall\ \lambda\in\mathbb{R},\ X\in V$, and with a base point $c\in V$ satisfying $N(c)=1$. Then, if the full linearization of the cubic norm, denoted by $N(X,Y,Z)$ and defined as

[TABLE]

is trilinear, the following four maps can be introduced :

The trace

[TABLE] 2. 2.

A quadratic map

[TABLE] 3. 3.

A bilinear map

[TABLE] 4. 4.

A trace bilinear form

[TABLE]

A cubic Jordan algebra $\mathfrak{J}$ with multiplicative identity $Id=c$ can be obtained starting from the vector space $V$ above iff $N$ is Jordan cubic, namely iff: I] The trace bilinear form (4.6) is non-degenerate, and II] the quadratic adjoint map, $\sharp\colon\mathfrak{J}\rightarrow\mathfrak{J}$, uniquely defined by $Tr(X^{\sharp},Y):=3N(X,X,Y)$, satisfies

[TABLE]

In a cubic Jordan algebra, the so-called Jordan product can be introduced in the following way :

[TABLE]

where $X\times Y$ denotes the linearization of the quadratic adjoint map :

[TABLE]

Another related map is the Jordan triple product :

[TABLE]

Jordan algebras were introduced and completely classified in [3] in an attempt to generalize quantum mechanics beyond the complex numbers $\mathbb{C}$. Below, we list all allowed possibilities of cubic Jordan algebras [4, 5, 9, 11, 13]:

the simplest case: $\mathfrak{J}$ $=\mathbb{R}$, $N(X)\doteq X^{3}$; 2. 2.

the infinite sequence of semi-simple Jordan algebras given by $\mathfrak{J}=\mathbb{R}\oplus\Gamma_{m,n}$ (named pseudo-Euclidean spin factors), where $\Gamma_{m,n}$ is an $\left(m+n\right)$-dimensional vector space over $\mathbb{R}$, namely the Clifford algebra of $O(m,n)$, with $N(X=\xi\oplus\gamma)\doteq\xi\gamma^{a}\gamma^{b}\eta_{ab}$; 3. 3.

Four exceptional, simple and Euclidean111The Lorentzian version of the exceptional, simple, Lorentzian cubic Jordan algebras can also be defined; for its definition and symmetries, see [36], and for its use in supergravity see [92]. cases, given by $\mathfrak{J}=J_{3}^{\mathbb{A}}$ or $\mathfrak{J}=J_{3}^{\mathbb{A}_{s}}$, the algebra of $3\times 3$ Hermitian matrices over the four division algebras $\mathbb{A=R}$ (real numbers)$,\mathbb{C}$ (complex numbers)$,\mathbb{H}$ (quaternions)$,\mathbb{O}$ (octonions), or their split versions $\mathbb{A}_{s}=\mathbb{C}_{s},\mathbb{H}_{s},\mathbb{O}_{s}$ :

[TABLE]

with conjugation (denoted by bar) pertaining to the relevant (division or split) algebra. In these cases, the cubic norm is given by

[TABLE]

This reproduces the usual determinant222For explicit constructions of $N(X)$, see e.g. [28] and [29]. for $\mathbb{A}=\mathbb{R\ }$ and $\mathbb{C}$. In these cases, the Jordan product simply reads

[TABLE]

where $XY$ is just the conventional $3\times 3$ matrix product; see e.g. [5] for a comprehensive account.

4.2 Jordan Pairs

Jordan algebras have traveled a long journey, since their appearance in the 30’s [3]. The modern formulation [80] involves a quadratic map $U_{x}y$ (like $xyx$ for associative algebras) instead of the original symmetric product (4.13). The quadratic map and its linearization $V_{x,y}z=(U_{x+z}-U_{x}-U_{z})y$ (like $xyz+zyx$ in the associative case) reveal the mathematical structure of Jordan algebras much more clearly, through the notion of inverse, inner ideal, generic norm, etc. The axioms are:

[TABLE]

The quadratic formulation led to the concept of Jordan triple systems [81], an example of which is a pair of modules represented by rectangular matrices. There is no way of multiplying two matrices $x$ and $y$ , say $n\times m$ and $m\times n$ respectively, by means of a bilinear product. But one can do it using a product like $xyx$, quadratic in $x$ and linear in $y$. Notice that, like in the case of rectangular matrices, there needs not be a unity in these structures. The axioms are in this case:

[TABLE]

Finally, a Jordan pair [14] is just a pair of modules $(V^{+},V^{-})$ acting on each other (but not on themselves) like a Jordan triple:

[TABLE]

where $\sigma=\pm$ and $x^{\sigma}\in V^{+\sigma}\,,\;y^{-\sigma}\in V^{-\sigma}$.

Jordan pairs [14] $(V^{+},V^{-})\doteq\left(\mathfrak{J},\mathfrak{J}^{\prime}\right)$ (whose recent mathematical and physical treatment can be found e.g. in [30, 31, 32]; also cfr. [12] for a review) are strictly related to the Tits-Kantor-Koecher construction of 3-graded Lie Algebras $\mathfrak{L}$ [15, 16, 17] (see also the interesting relation to Hopf algebras [18]):

[TABLE]

where $\mathfrak{J}$ is a Jordan algebra, $str(\mathfrak{J})=\mathcal{L}(\mathfrak{J})\oplus der(\mathfrak{J})$ is the structure Lie algebra of $\mathfrak{J}$ [12], $\mathcal{L}\left(\mathfrak{J}\right)$ is the left multiplication in $\mathfrak{J}$, and $der(\mathfrak{J})=[\mathcal{L}(\mathfrak{J}),\mathcal{L}(\mathfrak{J})]$ is the algebra of derivations of $\mathfrak{J}$ (i.e., the algebra of the automorphism group of $\mathfrak{J}$) [19, 20]. As we will see in the next subsection (cfr. (4.21)), $\mathfrak{L}$ can be identified with the conformal Lie algebra $conf(\mathfrak{J})$ of $\mathfrak{J}$ itself.

4.3 Symmetries of Jordan Algebras

To each cubic Jordan algebra, a number of symmetry groups can be associated :

• •

$Aut(\mathfrak{J})$, the group of automorphisms of $\mathfrak{J}$, which leaves invariant the structure constants of the Jordan product (the Lie algebra of $Aut(\mathfrak{J})$ is given by the derivations $der(\mathfrak{J})$ of $\mathfrak{J}$).

• •

$Str(\mathfrak{J})$, the structure group, with Lie algebra $str(\mathfrak{J})$, which leaves the cubic norm $N$ invariant up to a rescaling:

[TABLE]

the reduced structure group $Str_{0}(\mathfrak{J})$, with Lie algebra $str_{0}(\mathfrak{J})$, is obtained from $Str(\mathfrak{J})$ by modding it out by its center [19, 5, 21]:

[TABLE]

It should be here remarked that the structure group of $\mathfrak{J}$ is the automorphism group of the Jordan pair $(\mathfrak{J},\mathfrak{J}^{\prime})$ :

[TABLE]

• •

$Conf(\mathfrak{J})$, the conformal group, whose Lie algebra $conf(\mathfrak{J})$ can be given a 3-graded structure with respect to $str(\mathfrak{J})$ :

[TABLE]

The Tits-Kantor-Koecher construction [15, 16, 17] (4.17) of $conf(\mathfrak{J})\equiv\mathfrak{L}$ establishes a one-to-one mapping between the grade +1 subspace $g^{1}$ of $conf(\mathfrak{J})$ and the corresponding Jordan algebra $\mathfrak{J}$ : $g^{1}\Leftrightarrow\mathfrak{J}$. Every Lie algebra $\mathfrak{L}$ (4.17) admits a conjugation (involutive automorphism) under which the elements of the grade +1 subspace get mapped into the elements of the grade -1 subspace, and vice versa (grade-reversing nature of the involution); cfr. e.g. [82]. Remarkably, $Conf(\mathfrak{J})$ is isomorphic to the automorphism group of the (reduced) Freudenthal triple system defined over $\mathfrak{J}$; see below.

• •

$QConf(\mathfrak{J})$, the quasi-conformal group, with Lie algebra $qconf(\mathfrak{J})$, which can be defined by introducing Freudenthal triple systems and their further extension named extended Freudenthal triple system [33]; see below.

All symmetry groups of (simple and semi-simple) cubic Jordan algebras over $\mathbb{R}$ are listed333Besides cubic Jordan algebras and their symmetries, in the subsequent treatment we will also consider another remarkable Hermitian Jordan triple system, namely given by $2$-dimensional octonionic vectors, and denoted by $M_{2,1}\left(\mathbb{O}\right)$. Its relevant symmetries are $conf\left(M_{2,1}\left(\mathbb{O}\right)\right)\cong E_{6(-14)}$, and $str_{0}\left(M_{2,1}\left(\mathbb{O}\right)\right)\cong so(8,2)$ (cfr. [22, 23, 34, 82] and [35]). in Table 1.

Remarkably, the symmetries of cubic Jordan algebras over $J_{3}^{\mathbb{A}}$ and $J_{3}^{\mathbb{A}_{s}}$ respectively arrange as entries of the single-split and doubly-split Magic Squares of order three (reported in Tables 2 and 3), which are non-compact, real forms of the Freudenthal-Rozenfeld-Tits Magic Square of Lie algebras itself [24] (reported in Table 4); generally, $aut$, $str_{0}$, $conf$ and $qconf$ Lie algebras enter the first, second, third and fourth rows of the $4\times 4$ array of algebras constituting the Magic Square (cfr. [36] for a comprehensive review).

4.4 Freudenthal Triple Systems

Starting from a cubic Jordan algebra $\mathfrak{J}$, a (reduced [21]) Freudenthal triple system (FTS) is defined as the vector space

[TABLE]

An element $\mathbf{x}\in\mathbf{F}\mathfrak{(J)}$ can thus formally be written as a “$2\times 2$ matrix” :

[TABLE]

An FTS is endowed444It is worth remarking that all the other necessary definitions, such as the cubic and trace bilinear forms, are inherited from the underlying Jordan algebra $\mathfrak{J}$. with a non-degenerate bilinear antisymmetric quadratic form, a quartic form and a trilinear triple product [25, 21, 26, 27, 9] :

Quadratic form $\{\bullet,\mathbf{\bullet}\}$: $\mathbf{F}\mathfrak{(J)}\times\mathbf{F}\mathfrak{(J)}\rightarrow\mathbb{R}$, defined as

[TABLE]

[TABLE] 2. 2.

Quartic form $\Delta:\mathbf{F}\mathfrak{(J)}\rightarrow\mathbb{R}$, defined as

[TABLE]

[TABLE] 3. 3.

Triple product $T:\mathbf{F}\mathfrak{(J)}\times\mathbf{F}\mathfrak{(J)}\times\mathbf{F}\mathfrak{(J)}\rightarrow\mathbf{F}\mathfrak{(J)}$, defined as

[TABLE]

where $\Delta(\mathbf{x},\mathbf{y},\mathbf{w},\mathbf{z})$ is the full linearization of $\Delta(\mathbf{x})$, such that $\Delta(\mathbf{x},\mathbf{x},\mathbf{x},\mathbf{x})=\Delta(\mathbf{x})$.

The automorphism group $Aut(\mathbf{F}\mathfrak{(J)})$, with Lie algebra $aut(\mathbf{F}\mathfrak{(J)})$, is defined as the set of all invertible $\mathbb{R}$-linear transformations which leave both $\{\mathbf{x},\mathbf{y}\}$ and $\Delta(\mathbf{x},\mathbf{y},\mathbf{w},\mathbf{z})$ invariant [21].

It can be proved [39, 40, 41] that, as anticipated above :

[TABLE]

4.5 Extended Freudenthal Triple Systems

Every simple Lie algebra $g$ can be endowed with a $5$-grading, determined by one of its generators $\mathcal{G}\equiv so(1,1)$, with one-dimensional $\pm 2$-graded subspaces :

[TABLE]

where

[TABLE]

As firstly discussed in [33], a $5$-graded555For $sl(2)$, the $5$-grading degenerates into a $3$-grading. Lie algebra $g$ can geometrically be constructed as the quasi-conformal Lie algebra $qconf\left(\mathfrak{J}\right)$ over a vector space $\mathbf{EF}\left(\mathfrak{J}\right)$, named extended Freudenthal triple system (EFTS), which is coordinatized by $\mathcal{X}:=\left(\mathbf{x},\Phi\right)\in\mathbf{EF}\left(\mathfrak{J}\right)$, where $\mathbf{x}\in\mathbf{F}\left(\mathfrak{J}\right)$, and $\Phi$ is an extra real variable [33, 42]:

[TABLE]

Remarkably, a norm $\mathcal{N}:\mathbf{EF}\mathfrak{(J)}\rightarrow\mathbb{R}$ can be defined by using the quartic form $\Delta$ previously introduced in $\mathbf{F}\left(\mathfrak{J}\right)$, as follows666Since the image of $\mathcal{\Delta}$ in $\mathbf{F}\left(\mathfrak{J}\right)$ extends over the whole $\mathbb{R}$, for $\mathcal{\Delta}(\mathbf{x})<0$ the light-like condition $\mathcal{N}(\mathcal{X})=0$ in $\mathbf{EF}\left(\mathfrak{J}\right)$ does not yield real solutions for $\Phi$. However, as discussed in [33], this problem can be solved by complexifying the whole $\mathbf{EF}\left(\mathfrak{J}\right)$ (i.e., by considering $\mathbb{F}=\mathbb{C}$ as ground field), thus obtaining a realization of the complexified Lie algebra $\mathfrak{g}\left(\mathbb{C}\right)$ over $\left[\mathbf{EF}\left(\mathfrak{J}\right)\right]_{\mathbb{C}}$. :

[TABLE]

Furthermore, a “quartic distance” $d_{4}:\mathbf{EF}\mathfrak{(J)\times}\mathbf{EF}\mathfrak{(J)}\rightarrow\mathbb{R}$ between any two points $\mathcal{X}=\left(\mathbf{x},\Phi\right)$ and $\mathcal{Y}:=\left(\mathbf{y},\Psi\right)$ in $\mathbf{EF}\left(\mathfrak{J}\right)$ can be defined as

[TABLE]

such that $\mathcal{N}(\mathcal{X})=d_{4}(\mathcal{X},\mathcal{Y}=0)$.

Then, the quasi-conformal group $QConf\left(\mathfrak{J}\right)$ over $\mathbf{EF}\left(\mathfrak{J}\right)$, with Lie algebra $qconf\left(\mathfrak{J}\right)$, is defined as the set of all invertible $\mathbb{R}$-linear transformations which leave invariant the “quartic light-cone”, namely, the geometrical locus defined by [33]

[TABLE]

Thus, every $5$-graded Lie algebra $g$, geometrically realized as the quasi-conformal Lie algebra $qconf\left(\mathfrak{J}\right)$ over a vector space $\mathbf{EF}\left(\mathfrak{J}\right)$, admits a conformal invariant given by the norm $\mathcal{N}$ (4.33); see also [42].

4.6 Embeddings

In this Subsection we will present some general structures of embeddings involving the symmetry algebras of cubic Jordan algebras introduced above, along with some comments on their physical meaning and relevance.

We start and remark that $str_{0}$, $conf$ and $qconf$ Lie algebras can be interpreted as global, electric-magnetic duality ($U$-duality777Here $U$-duality is referred to as the “continuous” symmetries of [43]. Their discrete versions are the $U$-duality non-perturbative string theory symmetries introduced in [44].) symmetries of suitable theories of gravity (possibly with local supersymmetry) coupled to scalar fields and Abelian vectors, respectively in $D=5,4,3$ (Lorenzian-signed) space-time dimensions (cfr. e.g. [45], and Refs. therein). Such symmetries are non-linearly realized on the scalars, while vectors do sit in some linear representations of them (in $D=3$, they are completely dualized into scalar fields, as well).

As mentioned above, $aut$, $str_{0}$, $conf$ and $qconf$ Lie algebras of cubic (i.e. rank-3) Jordan algebras over $J_{3}^{\mathbb{A}}$ and $J_{3}^{\mathbb{A}_{s}}$ respectively enter the first, second, third and fourth rows of the single-split and doubly-split Magic Squares of order three (see related Tables). In general, such algebras are embedded maximally into the algebras of the row above, along the same column, possibly with a commuting summand; namely, the following maximal embeddings hold (cfr. e.g. [36], and Refs. therein):

[TABLE]

and these actually hold for any of the (simple and semi-simple) cubic Jordan algebras introduced above.

Within the physical interpretation of $U$-dualities, the $so(1,1)$ in (4.37) can be regarded as corresponding to the Kaluza-Klein (KK) compactification radius of the $S^{1}$-reduction from $D=5$ to $D=4$; alternatively, such an $so(1,1)$ can also be conceived as the Lie algebra associated to the pseudo-Kähler connection of the pseudo-special Kähler (and pseudo-Riemannian) symmetric coset $\frac{Conf}{Str_{0}\times SO(1,1)}$, obtained from888“$mcs$” denotes the maximal compact subalgebra/subgroup throughout. Note that $mcs\left(str_{0}\right)=aut$ for $\mathfrak{J}=J_{3}^{\mathbb{A}}$, $\mathbb{R}$ and $\mathbb{R}\oplus\Gamma_{m,n}$. $\frac{Str_{0}}{mcs\left(Str_{0}\right)}$ by applying the inverse $R^{\ast}$-map pertaining to a timelike compactification from $D=5$ Lorentzian dimensions to $D=4$ spacelike dimensions [46, 47]. On the other hand, the $sl(2,\mathbb{R})$ in (4.38) can be identified as corresponding to the Ehlers symmetry $sl(2,\mathbb{R})_{Ehlers}$ arising from the $S^{1}$-reduction from $D=4$ to $D=3$; such an $sl(2,\mathbb{R})$ can also be regarded as the Lie algebra associated to the connection of the para-quaternionic (and pseudo-Riemannian) symmetric coset $\frac{QConf}{Conf\times SL(2,\mathbb{R})}$, obtained from $\frac{Conf}{mcs\left(Conf\right)}$ by applying the inverse $c^{\ast}$-map pertaining to a timelike compactification from $D=4$ Lorentzian dimensions to $D=3$ spacelike dimensions999Note that the $\frac{Conf}{mcs\left(Conf\right)}$ is (special) Kähler only in certain cases. [48, 49, 47].

A generalization of the embedding (4.38) is provided by the so-called super-Ehlers embeddings, recently discussed in [50] :

[TABLE]

where $g_{3}$ is the $D=3$ $U$-duality Lie algebra (namely, $qconf$ in the cases treated above), $g_{D}$ is the $U$-duality Lie algebra in $3<D\leqslant 11$ dimensions, and $sl(D-2,\mathbb{R})_{Ehlers}$ is the Ehlers algebra in $D$ dimensions. These embeddings, discussed and generally proven in [50] (see also [51]) have different features, depending on $D$; they can be maximal or non-maximal, symmetric or non-symmetric, etc.. The (symmetric and maximal) case $D=4$ of (4.39), matching (4.38), is usually simply named Ehlers embedding. Morever, the (non-symmetric but maximal) case $D=5$ of (4.39) pertains to the Jordan pairs introduced above, and it is therefore usually dubbed Jordan-Pairs’ embedding.

As the embeddings (4.36)-(4.38) are obtained by moving along the columns of the relevant Magic Square (for a fixed row entry), another class of embeddings can be obtained by moving along the rows of the relevant Magic Square (for a fixed column entry). In symmetric Magic Squares, as the non-split ${\mathcal{L}}_{3}(\mathbb{A},\mathbb{B})$ and the double-split ${\mathcal{L}}_{3}(\mathbb{A}_{s},\mathbb{B}_{s})$ (respectively given in Table 4 and 3), these embeddings formally match (4.36)-(4.38), but their intepretation corresponds to the restriction from one (division $\mathbb{A}$ or split $\mathbb{A}_{s}$) algebra to the next smaller (division $\mathbb{A}$ or split $\mathbb{A}_{s}$) algebra. For the single-split Magic Square ${\mathcal{L}}_{3}(\mathbb{A}_{s},\mathbb{B})$ (reported in Table 2), this class presents different embeddings (at the level of non-compact, real forms) with respect to the ones given in (4.36)-(4.38). In general, it holds that (recall $aut_{\mathbb{B}}=der_{\mathbb{B}}$):

[TABLE]

where

[TABLE]

with $tri\mathbb{\ }$and $so$ respectively denote the triality and orthogonal (norm-preserving) Lie algebras (see e.g. [52, 53], and Refs. therein). More explicitly :

[TABLE]

In [52], the appearance of $\mathcal{A}_{q}$ was observed within the study of the charge orbits of asymptotically flat [math]- (black holes) and $1$- (black strings) branes in minimal “magical” Maxwell-Einstein supergravity theories in $D=5$ space-time dimensions. Moreover, it is worth noticing that $\mathcal{A}_{q}$ also occurs in the treatment of supergravity billiards and timelike Kaluza-Klein reductions (for recent treatment and set of related Refs., see e.g. [53]).

All in all, the right-hand side of (4.40) expresses the consequences of the algebraic embedding on its left-hand side at the level of $aut$, $str_{0}$, $conf$ and $qconf$ symmetries pertaining to the (division or split) algebras (as denoted by the subscripts). Clearly, one may consider non-maximal algebraic embeddings $\mathbb{B}\subset\mathbb{C}$, as well.

Another remarkable class of embedding involves the relation between simple cubic Jordan algebras $J_{3}^{\mathbb{A}}$ or $J_{3}^{\mathbb{A}_{s}}$ and some elements of the (bi-parametric) infinite sequence of semi-simple Jordan algebras $\mathbb{R}\oplus\mathbf{\Gamma}_{m,n}$ introduced above, exploiting the Jordan-algebraic isomorphisms

[TABLE]

where $q:=$dim${}_{R}\mathbb{A}=8,4,2,1$ for $\mathbb{A}=\mathbb{O},\mathbb{H},\mathbb{C},\mathbb{R}$, and $q:=$dim${}_{R}\mathbb{A}_{s}=8,4,2$ for $\mathbb{A}_{s}=\mathbb{O}_{s},\mathbb{H}_{s},\mathbb{C}_{s}$ (see e.g. App. A of [45] - and Refs. therein - for an introduction to division and split algebras). Indeed, the following (maximal, rank-preserving) Jordan-algebraic embeddings hold :

[TABLE]

Thus, one can consider their consequences at the level of symmetries of cubic Jordan algebras defined over the corresponding algebras, obtaining :

[TABLE]

It is here worth noticing the maximal nature of the embeddings (4.53), as well as the presence of the algebras $\mathcal{A}_{q}$ and $\widetilde{\mathcal{A}}_{q}$. Within the physical ($U$-duality) interpretation, $\mathcal{A}_{q}$ and $\widetilde{\mathcal{A}}_{q}$ are consistent with the properties of spinors in $q+2$ dimensions, with Lorentzian signature $\left(1,q+1\right)$ resp. Kleinian signature $\left(q/2+1,q/2+1\right)$; indeed, the electric-magnetic ($U$-duality) symmetry algebra in $D=6$ (Lorentzian) space-time dimensions is $so\left(1,q+1\right)\oplus\mathcal{A}_{q}$ for $\mathbb{A}$-based theories (which are endowed with minimal, chiral $\left(1,0\right)$ supersymmetry) and $so\left(q/2+1,q/2+1\right)\oplus\widetilde{\mathcal{A}}_{q}$ for $\mathbb{A}_{s}$-based theories (which are non-supersymmetric for $q=2,4$ - see [83] for a recent treatment - and endowed with maximal, non-chiral $\left(2,2\right)$ supersymmetry for $q=8$); cfr. e.g. [54], [55] (and Refs. therein) and [56] for further discussion.

In light of the above treatment, the Table of page 10 enjoys a rather simple Jordan algebraic interpretation, given in the Table at page 18, which thus characterizes the (maximal) parabolical relation among non-compact real forms of exceptional Lie algebras in terms of relations among Lie symmetries of rank-2 (i.e., quadratic) and rank-3 (i.e. cubic) Jordan algebras.

Table B : Jordan algebraic interpretation of maximally parabolically related non-compact real forms of finite-dimensional exceptional Lie algebras

${\cal G}$

${\cal M}$

${\cal G}^{\prime}$

${\cal M}^{\prime}$

${\rm dim}{\cal N}$

1 : $conf\left(M_{2,1}\left(\mathbb{O}\right)\right)$

$su(5,1)$

$qconf\left(J_{3}^{\mathbb{C}_{s}}\right)$

$conf\left(J_{3}^{\mathbb{C}_{s}}\right)$

$21$

$qconf\left(J_{3}^{\mathbb{C}}\right)$

$conf\left(J_{3}^{\mathbb{C}}\right)$

2 : $conf\left(M_{2,1}\left(\mathbb{O}\right)\right)$

$str_{0}\left(\Gamma_{7,1}\right)\oplus u(1)$

$qconf\left(J_{3}^{\mathbb{C}}\right)$

$str_{0}\left(\Gamma_{5,3}\right)\oplus u(1)$

$24$

3 : $str_{0}\left(J_{3}^{\mathbb{O}}\right)$

$str_{0}\left(J_{2}^{\mathbb{O}}\right)$

$str_{0}\left(J_{3}^{\mathbb{O}_{s}}\right)$

$str_{0}\left(J_{2}^{\mathbb{O}_{s}}\right)$

$16$

4 : $qconf\left(J_{3}^{\mathbb{C}_{s}}\right)$

$str_{0}\left(J_{3}^{\mathbb{C}_{s}}\right)\oplus sl(2,\mathbb{R})$

$qconf\left(J_{3}^{\mathbb{C}}\right)$

$str_{0}\left(J_{3}^{\mathbb{C}}\right)\oplus sl(2,\mathbb{R})$

$29$

5 : $conf\left(J_{3}^{\mathbb{O}}\right)$

$str_{0}\left(J_{3}^{\mathbb{O}}\right)$

$conf\left(J_{3}^{\mathbb{O}_{s}}\right)$

$str_{0}\left(J_{3}^{\mathbb{O}_{s}}\right)$

$27$

6 : $conf\left(J_{3}^{\mathbb{O}}\right)$

$str_{0}\left(J_{2}^{\mathbb{O}}\right)\oplus sl(2,\mathbb{R})$

$conf\left(J_{3}^{\mathbb{O}_{s}}\right)=qconf\left(J_{3}^{\mathbb{H}_{s}}\right)$

$str_{0}\left(J_{2}^{\mathbb{O}_{s}}\right)\oplus sl(2,\mathbb{R})$

$42$

$qconf\left(J_{3}^{\mathbb{H}}\right)$

$str_{0}\left(\Gamma_{7,3}\right)\oplus su(2)$

7 : $conf\left(J_{3}^{\mathbb{O}}\right)$

$conf\left(J_{2}^{\mathbb{O}}\right)$

$conf\left(J_{3}^{\mathbb{O}_{s}}\right)=qconf\left(J_{3}^{\mathbb{H}_{s}}\right)$

$conf\left(J_{2}^{\mathbb{O}_{s}}\right)=conf\left(J_{3}^{\mathbb{H}_{s}}\right)$

$33$

$qconf\left(J_{3}^{\mathbb{H}}\right)$

$conf\left(J_{3}^{\mathbb{H}}\right)$

8 : $qconf\left(J_{3}^{\mathbb{H}_{s}}\right)$

$sl(4,\mathbb{R})\oplus\widetilde{\mathcal{A}}_{4}\oplus sl(3,\mathbb{R})$

$qconf\left(J_{3}^{\mathbb{H}}\right)$

$su^{\ast}(4)\oplus\mathcal{A}_{4}\oplus sl(3,\mathbb{R})$

$53$

9 : $qconf\left(J_{3}^{\mathbb{H}}\right)$

$str_{0}\left(J_{3}^{\mathbb{H}}\right)\oplus sl(2,\mathbb{R})$

$qconf\left(J_{3}^{\mathbb{H}_{s}}\right)=conf\left(J_{3}^{\mathbb{O}_{s}}\right)$

$str_{0}\left(J_{3}^{\mathbb{H}_{s}}\right)\oplus sl(2,\mathbb{R})~{}$

$47$

$conf\left(J_{3}^{\mathbb{O}}\right)$

$str_{0}\left(J_{3}^{\mathbb{H}}\right)\oplus su(2)$

10 : $qconf\left(J_{3}^{\mathbb{O}}\right)$

$conf\left(J_{3}^{\mathbb{O}}\right)$

$qconf\left(J_{3}^{\mathbb{O}_{s}}\right)$

$conf\left(J_{3}^{\mathbb{O}_{s}}\right)$

$57$

11 : $qconf\left(J_{3}^{\mathbb{O}}\right)$

$qconf\left(\mathbb{R}\oplus\Gamma_{8,0}\right)$

$qconf\left(J_{3}^{\mathbb{O}_{s}}\right)$

$qconf\left(\mathbb{R}\oplus\Gamma_{4,4}\right)$

$78$

12 : $qconf\left(J_{3}^{\mathbb{O}}\right)$

$str_{0}\left(J_{3}^{\mathbb{O}}\right)\oplus sl(2,\mathbb{R})$

$qconf\left(J_{3}^{\mathbb{O}_{s}}\right)$

$str_{0}\left(J_{3}^{\mathbb{O}_{s}}\right)\oplus sl(2,\mathbb{R})$

$83$

13 : $qconf\left(J_{3}^{\mathbb{O}}\right)$

$str_{0}\left(J_{2}^{\mathbb{O}}\right)\oplus sl(3,\mathbb{R})$

$qconf\left(J_{3}^{\mathbb{O}_{s}}\right)$

$str_{0}\left(J_{2}^{\mathbb{O}_{s}}\right)\oplus sl(3,\mathbb{R})$

$97$

14 : $der\left(J_{1,2}^{\mathbb{O}}\right)$

$str_{0}\left(\Gamma_{7,0}\right)$

$der\left(J_{1,2}^{\mathbb{O}_{s}}\right)$

$str_{0}\left(\Gamma_{4,3}\right)$

$15$

Some comments are in order (for further details, see Sec. 5).

The maximal parabolical relation 1. hints for a quasi-conformal interpretation of $E_{6(-14)}$, despite it is characterized as conformal symmetry algebra of the Hermitian Jordan triple system $M_{2,1}(\mathbb{O})$ [82]. In fact, $E_{6(-14)}$ is the U-duality symmetry of $\mathcal{N}=10$, $D=3$ supergravity (after complete dualization of 1-forms), obtained as the dimensional reduction of $\mathcal{N}=5$, $D=4$ supergravity, which does not admit matter coupling, and whose U-duality algebra is $su(5,1)$. Usually, U-duality symmetries in $D=4$ and $D=3$ can be characterized as conformal resp. quasi-conformal symmetries of Jordan triple systems. Essentially, a quasi-conformal realization of $E_{6(-14)}$ would concern an EFTS of non-reduced type (i.e., whose corresponding FTS is not constructed in terms of cubic Jordan algebras). Work is in progress to investigate this possibility, which would render the maximal parabolical relation 1. simply the restriction from $qconf$ to $conf$ symmetries of $M_{2,1}(\mathbb{O})$, $J_{3}^{\mathbb{C}}$ and $J_{3}^{\mathbb{C}_{s}}$. 2. 2.

In the maximal parabolical relation 2., $str_{0}\left(\Gamma_{5,3}\right)$ can be enhanced to $str_{0}\left(\Gamma_{6,4}\right)=qconf\left(\mathbb{R}\oplus J_{2}^{\mathbb{C}}\right)$, thus allowing for the commuting algebra $u(1)$ to be interpreted as $\mathcal{A}_{2}$; thus, the parabolic relation between $E_{6(2)}$ and $so(5,3)\oplus u(1)$ can be interpreted as a consequence of the restriction, at $qconf$ level, of the simple cubic Jordan algebra $J_{3}^{\mathbb{C}}$ to its maximal semi-simple Jordan subalgebra $\mathbb{R}\oplus J_{2}^{\mathbb{C}}\simeq\mathbb{R}\oplus\Gamma_{3,1}$. Again, this would hint for a $qconf$ interpretation of $E_{6(-14)}$, despite the fact that $str_{0}\left(\Gamma_{7,1}\right)$ cannot be characterized as $qconf$ symmetry. 3. 3.

The maximal parabolical relation 3. can be simply explained as the realization, at $str_{0}$ level, of the Jordan algebraic restriction $J_{3}^{\mathbb{O}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}}$ and $J_{3}^{\mathbb{O}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}$, noting that $str_{0}\left(\mathbb{R}\oplus\Gamma_{m,n}\right)=so(1,1)\oplus str_{0}\left(\Gamma_{m,n}\right)$. 4. 4.

The maximal parabolical relation 4. can be simply explained as the realization, for $J_{3}^{\mathbb{C}_{s}}$ and $J_{3}^{\mathbb{C}}$, of the embedding $qconf\supset str_{0}\oplus sl(3,\mathbb{R})$ (Jordan pairs’ embedding, or $D=5$ Ehlers embedding), with the $sl(3,\mathbb{R})$ Ehlers symmetry further branched to $gl(2,\mathbb{R})$ in order to give rise to the grading $so(1,1)$ algebra. 5. 5.

The maximal parabolical relation 5. can be simply explained as the realization, for $J_{3}^{\mathbb{O}}$ and $J_{3}^{\mathbb{O}_{s}}$, of the embedding $conf\supset str_{0}\oplus so(1,1)$. 6. 6.

The maximal parabolical relation 6. is based on the two-fold characterization of $E_{7(7)}$ as $conf\left(J_{3}^{\mathbb{O}_{s}}\right)$ (thus allowing a parabolical relation to $conf\left(J_{3}^{\mathbb{O}}\right)$), as well as $qconf\left(J_{3}^{\mathbb{H}_{s}}\right)$ (thus allowing a relation to $qconf\left(J_{3}^{\mathbb{H}}\right)$). Concerning the $conf$-part of the parabolical relation, we note that $str_{0}\left(J_{2}^{\mathbb{O}}\right)\oplus sl(2,\mathbb{R})$ can be enhanced to $conf\left(J_{2}^{\mathbb{O}}\right)\oplus sl(2,\mathbb{R})\simeq conf\left(\mathbb{R}\oplus J_{2}^{\mathbb{O}}\right)$, and thus it can be traced back to the Jordan algebraic restriction $J_{3}^{\mathbb{O}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}}$ at $conf$ level. On the other hand, concerning the $qconf$-part of the parabolical relation, we note that $str_{0}\left(J_{2}^{\mathbb{O}_{s}}\right)\oplus sl(2,\mathbb{R})$ can be enhanced to $qconf\left(J_{2}^{\mathbb{H}_{s}}\right)\oplus sl(2,\mathbb{R})$, and thus it can be traced back to the non-maximal Jordan algebraic restriction $J_{3}^{\mathbb{H}_{s}}\supset J_{2}^{\mathbb{H}_{s}}$ at $qconf$ level, with the algebra $sl(2,\mathbb{R})$ interpreted as $\widetilde{\mathcal{A}}_{4}$. Analogously, $str_{0}\left(\Gamma_{7,3}\right)\oplus su(2)$ can be enhanced to $qconf\left(J_{2}^{\mathbb{H}}\right)\oplus su(2)$, and thus it can be traced back to the non-maximal Jordan algebraic restriction $J_{3}^{\mathbb{H}}\supset J_{2}^{\mathbb{H}}$ at $qconf$ level, with the algebra $su(2)$ interpreted as $\mathcal{A}_{4}$. 7. 7.

Also the maximal parabolical relation 7. is based on the two-fold characterization of $E_{7(7)}$ as $conf\left(J_{3}^{\mathbb{O}_{s}}\right)$ (thus allowing a parabolical relation to $conf\left(J_{3}^{\mathbb{O}}\right)$), as well as $qconf\left(J_{3}^{\mathbb{H}_{s}}\right)$ (thus allowing a relation to $qconf\left(J_{3}^{\mathbb{H}}\right)$). Concerning the $conf$-part of the parabolical relation, we note that $conf\left(J_{2}^{\mathbb{O}}\right)$ can be enhanced to $conf\left(J_{2}^{\mathbb{O}}\right)\oplus sl(2,\mathbb{R})\simeq conf\left(\mathbb{R}\oplus J_{2}^{\mathbb{O}}\right)$, and thus it can be traced back to the Jordan algebraic restriction $J_{3}^{\mathbb{O}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}}$ at $conf$ level. Analogously, we note that $conf\left(J_{2}^{\mathbb{O}_{s}}\right)$ can be enhanced to $conf\left(J_{2}^{\mathbb{O}_{s}}\right)\oplus sl(2,\mathbb{R})\simeq conf\left(\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}\right)$, and thus it can be traced back to the Jordan algebraic restriction $J_{3}^{\mathbb{O}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}$ at $conf$ level. On the other hand, concerning the $qconf$-part of the parabolical relation, it can be interpreted as the realization, for $J_{3}^{\mathbb{H}_{s}}$ and $J_{3}^{\mathbb{H}}$, of the embedding restriction $qconf\supset conf\oplus sl(2,\mathbb{R})$ ($D=4$ Ehlers embedding), with the $sl(2,\mathbb{R})$ Ehlers symmetry further branched to the grading $so(1,1)$ algebra. 8. 8.

The maximal parabolical relation 8. can be explained as the realization, for $J_{3}^{\mathbb{H}_{s}}$ and $J_{3}^{\mathbb{H}}$, of the embedding $qconf\supset str_{0}\oplus sl(3,\mathbb{R})$ (Jordan pairs’ embedding, or $D=5$ Ehlers embedding). Indeed, $sl(4,\mathbb{R})\oplus\widetilde{\mathcal{A}}_{4}$ and $su^{\ast}(4)\oplus\mathcal{A}_{4}$ can respectively be enhanced to $str_{0}\left(J_{3}^{\mathbb{H}_{s}}\right)$ and $str_{0}\left(J_{3}^{\mathbb{H}_{s}}\right)$. 9. 9.

The maximal parabolical relation 9. is based on the two-fold characterization of $E_{7(7)}$ as $conf\left(J_{3}^{\mathbb{O}_{s}}\right)$ (thus allowing a parabolical relation to $conf\left(J_{3}^{\mathbb{O}}\right)$), as well as $qconf\left(J_{3}^{\mathbb{H}_{s}}\right)$ (thus allowing a relation to $qconf\left(J_{3}^{\mathbb{H}}\right)$). Concerning the $qconf$-part of the parabolical relation, we note that $str_{0}\left(J_{3}^{\mathbb{H}}\right)\oplus sl(2,\mathbb{R})$ can be enhanced to $str_{0}\left(J_{3}^{\mathbb{H}}\right)\oplus sl(3,\mathbb{R})$ (Jordan pairs’ embedding). Analogously, $str_{0}\left(J_{3}^{\mathbb{H}_{s}}\right)\oplus sl(2,\mathbb{R})$ can be enhanced to $str_{0}\left(J_{3}^{\mathbb{H}_{s}}\right)\oplus sl(3,\mathbb{R})$. On the other hand, concerning the $conf$-part of the parabolical relation, we note that the relation between $E_{7(7)}$ and $sl(6,\mathbb{R})\oplus sl(2,\mathbb{R})$ can be interpreted as the realization, at $conf$ level, of the non-maximal algebraic restriction $\mathbb{O}_{s}\supset\mathbb{C}_{s}$; indeed, $sl(6,\mathbb{R})$ enjoys a twofold characterization : as $str_{0}\left(J_{3}^{\mathbb{H}_{s}}\right)$ and as $conf\left(J_{3}^{\mathbb{C}_{s}}\right)$. Finally, the relation between $E_{7(-25)}$ and $su^{\ast}(6)\oplus su(2)$ is less direct, in the sense that it moves before horizontally along the 3rd row of the single-split (non-symmetric) Magic Square $\mathcal{L}_{3}(\mathbb{A}_{s},\mathbb{B})$, from the slot $\mathbb{O}$ to the slot $\mathbb{H}$, and then it moves vertically along the 3rd column, from the slot $\mathbb{H}_{s}$ to $\mathbb{C}_{s}$ (note that $E_{7(-25)}$ does not have a $qconf$ interpretation). 10. 10.

The maximal parabolical relation 10. can be simply explained as the realization, for $J_{3}^{\mathbb{O}}$ and $J_{3}^{\mathbb{O}_{s}}$, of the embedding $qconf\supset conf\oplus sl(2,\mathbb{R})$ ($D=4$ Ehlers embedding). 11. 11.

The maximal parabolical relation 11. can be explained as the realization, at $qconf$ level, of the maximal Jordan algebraic embeddings $J_{3}^{\mathbb{O}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}}$ and $J_{3}^{\mathbb{O}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}$. Indeed, $qconf\left(\mathbb{R}\oplus\Gamma_{8,0}\right)$ and $qconf\left(\mathbb{R}\oplus\Gamma_{4,4}\right)$ can respectively be enhanced to $qconf\left(\mathbb{R}\oplus\Gamma_{9,1}\simeq\mathbb{R}\oplus J_{2}^{\mathbb{O}}\right)=$ and $qconf\left(\mathbb{R}\oplus\Gamma_{5,5}\simeq\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}\right)$. 12. 12.

The maximal parabolical relation 12. can be explained as the realization, for $J_{3}^{\mathbb{O}}$ and $J_{3}^{\mathbb{O}_{s}}$, of the embedding $qconf\supset str_{0}\oplus sl(3,\mathbb{R})$ (Jordan pairs’ embedding, or $D=5$ Ehlers embedding). Indeed, $str_{0}\left(J_{3}\right)\oplus sl(2,\mathbb{R})$ can be enhanced to $str_{0}\left(J_{3}\right)\oplus gl(2,\mathbb{R})$, which is maximal in $str_{0}(J_{3})\oplus sl(3,\mathbb{R})$. 13. 13.

The maximal parabolical relation 13. can also be explained as the realization, for $J_{3}^{\mathbb{O}}$ and $J_{3}^{\mathbb{O}_{s}}$, of the embedding $qconf\supset str_{0}\oplus sl(3,\mathbb{R})$ (Jordan pairs’ embedding, or $D=5$ Ehlers embedding). Indeed, $str_{0}(J_{2})\oplus sl(3,\mathbb{R})$ can be enhanced to $str_{0}(J_{3})\oplus sl(3,\mathbb{R})$. 14. 14.

The maximal parabolical relation 14. can be explained as the realization, for $J_{1,2}^{\mathbb{O}}$ and $J_{1,2}^{\mathbb{O}_{s}}$ (i.e. for $q=8$), respectively of the maximal (symmetric) embeddings

[TABLE]

Indeed, $str_{0}\left(\Gamma_{q-1,0}\right)$ and $str_{0}\left(\Gamma_{q/2,q/2-1}\right)$ can trivially be enhanced to $str_{0}\left(\Gamma_{q,1}\right)$ resp. $str_{0}\left(\Gamma_{q/2+1,q/2}\right)$, which in turn are maximal in $str_{0}\left(\Gamma_{q+1,1}\simeq J_{2}^{\mathbb{A}}\right)$ and in $str_{0}\left(\Gamma_{q/2+1,q/2+1}\simeq J_{2}^{\mathbb{A}_{s}}\right)$, respectively.

5 Jordan Structures in Maximal Parabolics of Exceptional Lie Algebras

: Analysis

In this Section, we will analyze all maximal parabolic subalgebras (shortened as parabolics) of all non-compact, real forms of finite-dimensional exceptional Lie algebras, determining them by means of (chains of) maximal embeddings, and providing Jordan algebraic interpretations for them, in light of the treatment given in previous  Section. We will also provide an $\mathcal{M}^{\max}$-covariant decomposition of the (generally reducible) vector spaces $\mathcal{N}^{\max}$’s, occurring in the Bruhat branchings giving rise to the maximal parabolics. The numbering of maximal parabolics refers to the listing of Sec. 3.

5.1 $E_{6(6)}$

This is the maximally non-compact (split) real form of $E_{6}$. Its Jordan interpretation is essentially twofold (due to the symmetry of the double-split Magic Square ${\mathcal{L}}_{3}(\mathbb{A}_{s},\mathbb{B}_{s})$ [37], reported in Table 3) :

[TABLE]

5.1.1 $\mathcal{P}_{1}^{6(6)}\cong\mathcal{P}_{5}^{6(6)}$

The maximal parabolics   $\mathcal{P}_{1}^{6(6)}\cong\mathcal{P}_{5}^{6(6)}$   from (3.2) corresponds to the Bruhat decomposition (2.5):

[TABLE]

yielding a $3$-grading101010The subscripts denote $so(1,1)$-weights throughout. Unless otherwise indicated, all embeddings are symmetric; non-symmetric embeddings will be denoted by a “$ns$” upperscript. Only (chains of) maximal embeddings are considered throughout.. Since $so(5,5)\cong str_{0}\left(J_{2}^{\mathbb{O}_{s}}\right)$, at least two Jordan algebraic interpretations (denoted by $I$ and $II$) of (5.3)-(5.4) can be given:

the first one is

[TABLE]

where the $so(1,1)$ generating the $3$-grading is the Kaluza-Klein (KK) $so(1,1)$ of the $S^{1}$-reduction $D=6\rightarrow 5$. 2. 2.

the second one stems from the Jordan algebraic embedding $J_{3}^{\mathbb{O}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}$, at the level of $str_{0}$ :

[TABLE]

where the $so(1,1)$ generating the $3$-grading is the dilatonic $so(1,1)$ in $D=5$.

5.1.2 ${\cal P}_{2}^{6(6)}$

The maximal parabolics  ${\cal P}_{2}^{6(6)}$  from (3.2) corresponds to the Bruhat decomposition (2.5):

[TABLE]

which can be obtained by the following chain of embeddings :

[TABLE]

exhibiting a $5$-grading of contact, with  $\mathcal{N}_{2}^{6(6)\pm}=\mathbf{1}_{\pm 2}+\mathbf{20}_{\pm 1}\,$.

Since $sl(6,\mathbb{R})$ has a two-fold Jordan algebraic interpretation :

[TABLE]

the first step of the above chain can be interpreted in two ways :

[TABLE]

namely the $D=4$ Ehlers embedding for $J_{3}^{\mathbb{C}_{s}}$. 2. 2.

as a consequence of the split algebraic embedding $\mathbb{O}_{s}\supset\mathbb{H}_{s}\Rightarrow\mathfrak{J}_{3}^{\mathbb{O}_{s}}\supset\mathfrak{J}_{3}^{\mathbb{H}_{s}}$, at the level of $str_{0}$ it holds that :

[TABLE]

5.1.3 ${\cal P}_{3}^{6(6)}\cong{\cal P}_{6}^{6(6)}$

The maximal parabolics   ${\cal P}_{3}^{6(6)}\cong{\cal P}_{6}^{6(6)}$   from (3.2) corresponds to the Bruhat decomposition (2.5):

[TABLE]

thus yielding a $5$-grading (recall that ${\rm dim}\mathcal{N}_{3}^{6(6)\pm}=25$), which can be obtained by the following chain of embeddings:

[TABLE]

The first embedding is the very same of the chain (5.8); here the $sl(6,\mathbb{R})$ is branched to generate the $so(1,1)$ producing the parabolic $5$-grading, whereas in (5.8) it is the branching of the $sl(2,\mathbb{R})$ algebra to produce the relevant $so(1,1)$.

5.1.4 ${\cal P}_{4}^{6(6)}$

The maximal parabolics  ${\cal P}_{4}^{6(6)}$   from (3.2) corresponds to the Bruhat decomposition (2.5):

[TABLE]

which can be obtained at least in two different ways, associated to two embedding chains111111For a recent quantum informational interpretation of the first step of the chain, see e.g. [84]., respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

(LABEL:7)-(5.18) and (5.19)-(5.20) both yield a $7$-grading with the same $\mathcal{M}_{4}^{6(6)}$ but with different $\mathcal{N}$’s :

(LABEL:7)-(5.18) implies $\mathcal{N}_{4}^{6(6)+}=\left(\mathbf{1,1,2}\right)_{3}+\left(\mathbf{3,3,1}\right)_{2}+\left(\mathbf{3}^{\prime}\mathbf{,3}^{\prime}\mathbf{,2}\right)_{1}$; 2. 2.

(5.19)-(5.20) yields $\mathcal{N}_{4}^{6(6)+}=\left(\mathbf{1,1,2}\right)_{3}+\left(\mathbf{3,3}^{\prime}\mathbf{,1}\right)_{2}+\left(\mathbf{3}^{\prime}\mathbf{,3,2}\right)_{1}$.

Namely, $1\leftrightarrow 2$ iff $\mathbf{3}^{\prime}\leftrightarrow\mathbf{3}$ in the second $sl(3,\mathbb{R})$ algebra of $\mathcal{M}_{4}^{6(6)}$.

The Jordan-algebraic interpretation of the first step of chains (LABEL:7)-(5.18) and (5.19)-(5.20) is at least twofold. Indeed, it holds that :

[TABLE]

A first interpretation (denoted by $I$) is provided by the Jordan pairs’ (JP) embedding for the $\mathbb{C}_{s}$-based gravity theories [83] :

[TABLE]

where the $sl(3,\mathbb{R})$ commuting factor is the Ehlers group in $D=5$ (Lorentzian-signed) space-time dimensions. A second interpretation (denoted by $II$) is based on the non-maximal split algebraic embedding $\mathbb{O}_{s}\supset\mathbb{C}_{s}\Rightarrow J_{3}^{\mathbb{O}_{s}}\supset J_{3}^{\mathbb{C}_{s}}$, at the level of $str_{0}$ :

[TABLE]

5.2 $E_{6(2)}$

The Jordan algebraic interpretation of $E_{6(2)}$ is

[TABLE]

5.2.1 ${\cal P}_{1}^{6(2)}$

The maximal parabolics  ${\cal P}_{1}^{6(2)}$   from (3.7) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the following embedding chain:

[TABLE]

and accounting the action of  ${\cal M}_{1}^{6(2)}$   we obtain a $5$-grading121212The first subscripts denote $so(2)$ charges. Note that we use Slansky’s conventions on the triality of $so(8)$ [85]., with $\mathcal{N}_{1}^{6(2)\pm}=\mathbf{8}_{v,0,\pm 2}+\mathbf{8}_{c,-3,\pm 1}+\mathbf{8}_{s,3,\pm 1}$.

The Jordan algebraic interpretation of the first embedding of (5.26)-(5.27) stems from the Jordan algebraic embedding $\mathfrak{J}_{3}^{\mathbb{C}}\supset\mathbb{R}\oplus\mathfrak{J}_{2}^{\mathbb{C}}$, at the $qconf$ level :

[TABLE]

5.2.2 ${\cal P}_{2}^{6(2)}$

The maximal parabolics  ${\cal P}_{2}^{6(2)}$   from (3.7) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained by the following embedding chain:

[TABLE]

thus yielding $9$-grading, noting that $\mathcal{N}_{2}^{6(2)\oplus}=\left(\mathbf{1,2,1}\right)_{-3,3}\oplus\left(\mathbf{1,1,2}\right)_{-3,3}\oplus\left(\mathbf{3,2,1}\right)_{3,1}\oplus\left(\mathbf{3,1,2}\right)_{-3,1}\oplus\left(\mathbf{3,1,1}\right)_{0,4}\oplus\left(\mathbf{3}^{\prime}\mathbf{,2,2}\right)_{0,2}\,$. It holds that :

[TABLE]

and a possible interpretation of the first step of the chain (5.30)-(5.31) is provided by the JP embedding for the $\mathbb{C}$-based gravity theories :

[TABLE]

where, once again, the $sl(3,\mathbb{R})$ commuting factor is the Ehlers group in $D=5$ (Lorentzian-signed) space-time dimensions. Also the second step of the chain (5.30)-(5.31) can be given a Jordan-algebraic interpretation; this latter stems from the same embedding of (5.28), but here considered at the level of the reduced structure symmetries :

[TABLE]

5.2.3 ${\cal P}_{3}^{6(2)}$

The maximal parabolics  ${\cal P}_{3}^{6(2)}$   from (3.7) corresponds to the Bruhat decomposition:

[TABLE]

which can be realized by at least two chains of embeddings, respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

(5.36)-(5.40) and (5.41)-(5.45) both yield a $7$-grading, but with different $\mathcal{N}$’s (both of real dimension $29$) :

(5.36)-(5.40) implies $\mathcal{N}_{3}^{6(2)+}=\left(\mathbf{1},\mathbf{1,2}\right)_{3}+\left(\mathbf{3},\mathbf{3}^{\prime}\mathbf{,1}\right)_{2}+\left(\mathbf{3}^{\prime},\mathbf{3,2}\right)_{1}$; 2. 2.

(5.41)-(5.45) yields $\mathcal{N}_{3}^{6(2)+}=\left(\mathbf{1,1,2}\right)_{3}+\left(\mathbf{3}^{\prime},\mathbf{3,1}\right)_{2}+\left(\mathbf{3},\mathbf{3}^{\prime}\mathbf{,2}\right)_{1}$.

Namely, $1\leftrightarrow 2$ iff $\mathbf{3}^{\prime}\leftrightarrow\mathbf{3}$ in the $sl(3,\mathbb{C})_{\mathbb{R}}$ algebra of $\mathcal{M}_{3}^{6(2)}$.

Both steps of each of the chains $1$ and $2$ admits at least one Jordan algebraic interpretation, as follows : the chain $1$ (5.36)-(5.40) starts with the so-called Ehlers embedding for $\mathbb{C}$-based magic supergravity theories, and then proceeds with an inverse $R^{\ast}$-map (generating an $so(1,1)_{KK}$, determining the $7$-grading):

[TABLE]

On the other hand, the chain $2$ (5.41)-(5.45) starts with the JP embedding for the $\mathbb{C}$-based magic supergravity theories (generating the131313The $D=5$ Ehlers $sl(3,\mathbb{R})_{Ehlers}$ can also be regarded as the enhancement of $sl(2,\mathbb{R})_{Ehlers}\oplus so(1,1)_{KK}$, obtained by a composition of an inverse $c^{\ast}$-map and of an inverse $R^{\ast}$-map. $D=5$ Ehlers $sl(3,\mathbb{R})_{Ehlers}$), and then proceeds with a further branching of this latter symmetry into $sl(2,\mathbb{R})_{Ehlers}$ ($D=4$ Ehlers) $\oplus\ so(1,1)_{KK}$ :

[TABLE]

5.2.4 ${\cal P}_{4}^{6(2)}$

The maximal parabolics  ${\cal P}_{4}^{6(2)}$   from (3.7) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained by the following embedding chain:

[TABLE]

exhibiting a $5$-grading with  $\mathcal{N}_{4}^{6(2)\pm}=\mathbf{1}_{\pm 2}+\mathbf{20}_{\pm 1}\,$.

The Jordan algebraic interpretation starts with the Ehlers embedding for $\mathbb{C}$-based theories, and then proceeds by further branching this latter symmetry, in order to generate the $so(1,1)$ responsible for the $5$-grading :

[TABLE]

Notice that in this case the $so(1,1)$ has not a KK interpretation, but it is rather the non-compact Cartan of $sl(2,\mathbb{R})_{Ehlers}$.

5.3 $E_{6(-14)}$

The Jordan interpretation of $E_{6(-14)}$ is twofold :

[TABLE]

where, as mentioned in the previous Section, $M_{1,2}\left(\mathbb{O}\right)$ denotes an Hermitian Jordan triple system formed by octonionic $2$-vectors (cfr. [22, 23, 34] and [35]). On the other hand, $\mathcal{K}\left(J_{3}^{\mathbb{O}}\right)$ stands for the stabilizer of the rank-$4$ orbit of the action of $Conf\left(J_{3}^{\mathbb{O}}\right)\cong Aut\left(\mathbf{F}\left(J_{3}^{\mathbb{O}}\right)\right)$ on its (fundamental) irrep. $\mathbf{56}$ with positive quartic invariant $I_{4}>0$ and representative “$++--$” (for further detail, cfr. e.g. [86, 87, 88]).

5.3.1 ${\cal P}_{1}^{6(-14)}$

The maximal parabolics  ${\cal P}_{1}^{6(-14)}$   from (3.12) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained by the following embedding chain:

[TABLE]

thus yielding a $5$-grading with  $\mathcal{N}_{1}^{6(-14)+}=\mathbf{8}_{v,0,2}+\mathbf{8}_{c,-3,1}+\mathbf{8}_{s,3,1}\,$.

A Jordan algebraic interpretation (of the first step) of the chain (5.55)-(5.56) is a consequence of the Jordan algebraic embedding $J_{3}^{\mathbb{O}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}}$ at the level of the $\mathcal{K}$-symmetry, namely :

[TABLE]

where $\mathcal{K}\left(\mathbb{R}\oplus J_{2}^{\mathbb{O}}\right)$ denotes stabilizer of the rank-$4$ orbit of the action of $Conf\left(\mathbb{R}\oplus J_{2}^{\mathbb{O}}\right)\cong Aut\left(\mathbf{F}\left(\mathbb{R}\oplus J_{2}^{\mathbb{O}}\right)\right)$ on its (bi-fundamental) rep. $\left(\mathbf{2},\mathbf{10}\right)$ with positive quartic invariant $I_{4}>0$ and representative “$++--$” [86, 87, 88]). Indeed, it holds that (cfr. e.g. case $4b$ with $n=10$ in Table VIII of [88])

[TABLE]

which is nothing but the $q=8$ case of the general relation :

[TABLE]

5.3.2 ${\cal P}_{2}^{6(-14)}$

The maximal parabolics  ${\cal P}_{2}^{6(-14)}$   from (3.12) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the following embedding chain:

[TABLE]

exhibiting a $5$-grading with  $\mathcal{N}_{2}^{6(-14)\pm}=\mathbf{1}_{\pm 2}+\mathbf{20}_{\pm 1}\,$.

A possible Jordan algebraic interpretation (of the first step) of the chain (5.61)-(5.62) would be based on the characterization of $su(5,1)$, which, at the moment, we can characterize as a maximal subalgebra (with commutant $sl(2,\mathbb{R})$) of $E_{6(-14)}$, only. As mentioned above,work is in progress for a sharper interpretation of $su(5,1)$.

5.4 $E_{6(-26)}$

This is the minimally non-compact, real form of $E_{6}$. Its Jordan interpretation reads:

[TABLE]

There is only one maximal parabolics  ${\cal P}^{6(-26)}$   from (3.15) corresponding to the Bruhat decomposition:

[TABLE]

thus yielding a $3$-grading. Since

[TABLE]

at least two Jordan algebraic interpretations (denoted by $1$ and $2$) of (5.64) can be given :

[TABLE]

where the $so(1,1)$ generating the $3$-grading is the KK $so(1,1)$ of the $S^{1}$-reduction $D=6\rightarrow 5$. 2. 2.

The second interpretation stems from the embedding $\mathfrak{J}_{3}^{\mathbb{O}}\supset\mathbb{R}\oplus\mathfrak{J}_{2}^{\mathbb{O}}$, evaluated at $str_{0}$ level :

[TABLE]

where the $so(1,1)$ generating the $3$-grading is the dilatonic $so(1,1)$ in $D=5$.

5.5 $E_{7(7)}$

This is the split real form of $E_{7}$. Its Jordan interpretation is essentially twofold (due to the symmetry of the double-split Magic Square ${\mathcal{L}}_{3}(\mathbb{A}_{s},\mathbb{B}_{s})$ [37], reported in Table 3) :

[TABLE]

5.5.1 ${\cal P}_{1}^{7(7)}$

The maximal parabolics  ${\cal P}_{1}^{7(7)}$   from (3.17) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the following embedding chain :

[TABLE]

thus yielding a $5$-grading as $\mathcal{N}_{1}^{7(7)+}=\mathbf{1}_{2}+\mathbf{32}_{1}^{\prime}$.

The Jordan algebraic interpretation of (the first step of) (5.71)-(5.72) is at least threefold, and it is based on the following identifications :

[TABLE]

The first interpretation stems from the embedding $\mathbb{O}_{s}\supset\mathbb{H}_{s}$, at $conf$ level :

[TABLE] 2. 2.

the second one stems from the embedding $J_{3}^{\mathbb{O}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}$ at the $conf$ level :

[TABLE] 3. 3.

the third interpretation stems from $J_{3}^{\mathbb{H}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{H}_{s}}$, at the $qconf$ level :

[TABLE]

5.5.2 ${\cal P}_{2}^{7(7)}$

The maximal parabolics  ${\cal P}_{2}^{7(7)}$   from (3.17) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two embedding chains, respectively denoted by $I$ and $II$ :

[TABLE]

[TABLE]

The chains of embeddings $I$ and $II$ give rise to a $7$-grading with $\mathcal{N}_{2}^{7(7)+}=\left(\mathbf{1},\mathbf{2}\right)_{3}+\left(\mathbf{15,1}\right)_{2}+\left(\mathbf{15}^{\prime},\mathbf{2}\right)_{1}\,$. It is here worth remarking that the second steps of the chain $I$ do pertain to two different, inequivalent (maximal, symmetric) embeddings of $gl(6,\mathbb{R})$ into $so(6,6)$, respectively denoted by $i$ and $ii$; such two embeddings can e.g. be discriminated by the branching of the chiral spinor irreps. $\mathbf{32}$ and $\mathbf{32}^{\prime}$ of $so(6,6)$, namely141414It is amusing to note that the branching $ii$ is overlooked in the otherwise fairly comprehensive treatment e.g. of [85] and [89]; it is however considered e.g. in [90].:

[TABLE]

Let us now consider the Jordan algebraic interpretation of the various chains. We start by observing that

[TABLE]

By also recalling (5.73), (5.75) and (5.2), possible interpretations read as follows:

[TABLE]

[TABLE]

5.5.3 ${\cal P}_{3}^{7(7)}$

The maximal parabolics  ${\cal P}_{3}^{7(7)}$   from (3.17) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained by at least two chains of embeddings151515We use the conventions of [91]., respectively denoted by $1$ and161616In the second step of chain $2$ there is complete symmetry between the two $so(3,3)$ factors, thus the choice of which one is branched in the third step is immaterial. $2$ (recall that $so(3,3)\cong sl(4,\mathbb{R})$) :

[TABLE]

[TABLE]

These chains of embeddings give rise to a $9$-grading, with $\mathcal{N}_{3\,;\,a)}^{7(7)+}\cong\mathcal{N}_{3\,;\,b)}^{7(7)+}=\left(\mathbf{1,3},\mathbf{1}\right)_{4}\oplus\left(\mathbf{4,1,2}\right)_{3}\oplus\left(\mathbf{6,3}^{\prime},\mathbf{1}\right)_{2}\oplus\left(\mathbf{4}^{\prime}\mathbf{,3},\mathbf{2}\right)_{1}\,$.

A Jordan algebraic interpretation of the chains $1$ and $2$ is based on the following identification :

[TABLE]

and it reads

[TABLE]

[TABLE]

The second step of the Jordan algebraic interpretation (5.104) of the chain $2$ is intentionally left explicit : intriguingly, it provides an enhancement of the symmetry obtained by the JP embedding for the $\left(\mathbb{R\oplus}J_{2}^{\mathbb{H}_{s}}\right)$-based theory : indeed, this latter reads

[TABLE]

and then the following enhancement takes place:

[TABLE]

where the $so(3,3)$ on the right-hand side is one of the two summands in the second step of the chain $2$. The enhancement (5.106) actually hints for another interpretation of the non-symmetric embedding

[TABLE]

where the subscript “$nm$” denotes its next-to-maximal nature (i.e., the fact that it is realized by two subsequent maximal embeddings, namely the first two steps of chain $2$). In fact, (5.107) can be interpreted as the $D=6$ case of the Ehlers embedding for the non-supersymmetric $\mathbb{H}_{s}$-based gravity theory (see [83] and Refs. therein), and its next-to-maximal nature is consistent with the treatment of [50]; in other words, (5.107) can be interpreted as follows:

[TABLE]

where therefore one of the two $so(3,3)$ factors (say, $so(3,3)_{\mathbf{I}}$) is conceived as $str_{0}\left(J_{2}^{\mathbb{H}_{s}}\right)$, while the other one (say, $so(3,3)_{\mathbf{II}}$) is nothing but the $D=6$ Ehlers symmetry, and the commuting $sl(2,\mathbb{R})$ factor is seen as $\widetilde{\mathcal{A}}_{4}$. As a consequence of this interpretation, (5.104) can concisely be rewritten as :

[TABLE]

where

[TABLE]

5.5.4 ${\cal P}_{4}^{7(7)}$

The maximal parabolics  ${\cal P}_{4}^{7(7)}$   from (3.17) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two chains of embeddings, respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

Both chains $1$ and $2$ give rise to a $7$-grading, with $\mathcal{N}_{4\,;\,a)}^{7(7)+}=\left(\mathbf{5,1}\right)_{6}\oplus\left(\mathbf{5}^{\prime}\mathbf{,3}\right)_{4}\oplus\left(\mathbf{10,3}^{\prime}\right)_{2}$ and $\mathcal{N}_{4\,;\,b)}^{7(7)+}=(\mathbf{5}^{\prime},\mathbf{1})_{12}\oplus(\mathbf{5},\mathbf{3}^{\prime})_{8}\oplus(\mathbf{10}^{\prime},\mathbf{3})_{4}$ (both of real dimension $50$). Note that $1\leftrightarrow 2)$ iff the weights of the parabolic $so(1,1)_{1}$ gets doubled, and iff $\mathbf{5}\leftrightarrow\mathbf{5}^{\prime}$ (yielding $\mathbf{10}\leftrightarrow\mathbf{10}^{\prime}$) and $\mathbf{3}\leftrightarrow\mathbf{3}^{\prime}$ in $sl(5,\mathbb{R})$ and $sl(3,\mathbb{R})$, respectively. A Jordan algebraic interpretation of the first step of chains $1$ and $2$ coincides with the one of chain $1$ of ${\cal P}_{3}^{7(7)}$, and of chain $1$ of case ${\cal P}_{2}^{7(7)}$, respectively.

5.5.5 ${\cal P}_{5}^{7(7)}$

The maximal parabolics  ${\cal P}_{5}^{7(7)}$   from (3.17) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain :

[TABLE]

giving rise to a $5$-grading, with $\mathcal{N}_{5}^{7(7)}=(\mathbf{10},\mathbf{1})_{2}\oplus\left(\mathbf{16}^{\prime}\mathbf{,2}\right)_{1}\,$. Since

[TABLE]

at least two Jordan algebraic interpretations (denoted by $I$ and $II$) of the chain above can be given, namely :

[TABLE]

[TABLE]

5.5.6 ${\cal P}_{6}^{7(7)}$

The maximal parabolics  ${\cal P}_{6}^{7(7)}$   from (3.17) corresponds to the Bruhat decomposition:

[TABLE]

giving rise to a $3$-grading. At least two Jordan algebraic interpretations (denoted by $I$ and $II$) of the chain above can be given :

[TABLE]

[TABLE]

It is worth remarking the different interpretation of the parabolic $so(1,1)$ (giving rise to the $3$-grading) in $I$ and $II$ : in $I$, it is identified with $\widetilde{\mathcal{A}}_{2}$, whereas in $II$ it is the $so(1,1)_{KK}$ of the $S^{1}$-reduction $D=5\rightarrow 4$.

5.5.7 ${\cal P}_{7}^{7(7)}$

The maximal parabolics  ${\cal P}_{7}^{7(7)}$   from (3.17) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

Thus, it gives rise to a $5$-grading, with $\mathcal{N}_{7}^{7(7)+}=\mathbf{35}_{4}^{\prime}\oplus\mathbf{7}_{8}\,$.

At least two Jordan algebraic interpretations (denoted by $I$ and $II$) of the first step of the chain above can be given, respectively pertaining to $\mathbb{H}_{s}$ and $\mathbb{O}_{s}$:

[TABLE]

[TABLE]

5.6 $E_{7(-5)}$

The Jordan interpretation of $E_{7(-5)}$ reads:

[TABLE]

5.6.1 ${\cal P}_{1}^{7(-5)}$

The maximal parabolics  ${\cal P}_{1}^{7(-5)}$   from (3.25) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

thus giving rise to a $5$-grading of contact type, with $\mathcal{N}_{1}^{7(-5)+}=\mathbf{1}_{2}\oplus\mathbf{32}_{1}^{\prime}$. By observing that

[TABLE]

a Jordan algebraic interpretation (of the first step) of the chain above is given by the Ehlers embedding for the $\mathbb{H}$-based magic supergravity theory :

[TABLE]

thus, the parabolic $so(1,1)$ (determining the $5$-grading) is the non-compact Cartan of the $D=4$ Ehlers $sl(2,\mathbb{R})_{Ehlers}$.

5.6.2 ${\cal P}_{2}^{7(-5)}$

The maximal parabolics  ${\cal P}_{2}^{7(-5)}$   from (3.25) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

Thus, it gives rise to a $5$-grading (cfr. (5.117)-(5.5.5)), with $\mathcal{N}_{5}^{7(-5)+}=(\mathbf{10},\mathbf{1})_{2}\oplus\left(\mathbf{16}^{\prime}\mathbf{,2}\right)_{1}\,$. Since

[TABLE]

a Jordan algebraic interpretation (of the first step) of the chain above is given by the embedding $J_{3}^{\mathbb{H}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{H}}$ considered at $qconf$ level :

[TABLE]

5.6.3 ${\cal P}_{3}^{7(-5)}$

The maximal parabolics  ${\cal P}_{3}^{7(-5)}$   from (3.25) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained at least through two chains of embeddings, respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

Both chains $1$ and $2$ give rise to a $7$-grading, with and $\ \mathcal{N}_{3}^{7(-5)+}=\left(\mathbf{1},\mathbf{2}\right)_{3}\oplus\left(\mathbf{15,1}\right)_{2}\oplus\left(\mathbf{15}^{\prime},\mathbf{2}\right)_{1}\,$. Once again, it is here worth remarking that the second step of the chain $2$ do pertain to two different, inequivalent (maximal, symmetric) embeddings of $su^{\ast}(6,\mathbb{R})\times so(1,1)$ into $so^{\ast}(12)$, respectively denoted by $i$ and $ii$; such two embeddings can e.g. be discriminated by (a different non-compact, real form of) the branching of the chiral spinor irreps. $\mathbf{32}$ and $\mathbf{32}^{\prime}$ of $so^{\ast}(12)$, cfr. (5.90)-(5.93).

Let us now consider the Jordan algebraic interpretation of the various chains. We observe that

[TABLE]

Thus, Jordan interpretations can be given as follows:

[TABLE]

[TABLE]

It is here worth commenting that the first step of the interpretation of $1$ is the JP* *embedding for the $\mathbb{H}$-based theory (determining the $D=5$ Ehlers $sl(3,\mathbb{R})_{Ehlers}$), and the resulting parabolic $so(1,1)$ (generating the $7$-grading) is the $so(1,1)$ commuting factor in the right-hand side of the maximal, symmetric embedding $sl_{Ehlers}(3,\mathbb{R})\supset sl(2,\mathbb{R})\oplus so(1,1)$. On the other hand, the first step of the interpretation of $2$ is the Ehlers embedding for the $\mathbb{H}$-based theory (determining the $D=4$ Ehlers $sl(2,\mathbb{R})_{Ehlers}$ through the inverse $c^{\ast}$-map [48, 49]), and the second step consists in an inverse $R^{\ast}$-map [46], which thus introduces the $so(1,1)_{KK}$ of the $S^{1}$-reduction $D=5\rightarrow 4$; in this case, this latter is the parabolic $so(1,1)$ (which generates the $7$-grading in $c)$).

5.6.4 ${\cal P}_{4}^{7(-5)}$

The maximal parabolics  ${\cal P}_{4}^{7(-5)}$   from (3.25) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two chains of embeddings, respectively denoted by $1$ and $2$ (recall that $su^{\ast}(4)\cong so(5,1)$, $su^{\ast}(2)\cong su(2)$) :

[TABLE]

[TABLE]

Note that $1$ and $2$ are different non-compact real forms of the chains $1$ and $2$ pertaining to ${\cal P}_{3}^{7(7)}$** **(and the same holds for the maximal parabolics under consideration). The chains of embeddings $1$ and $2$ give rise to a $7$-grading, with $\mathcal{N}_{4}^{7(-5)+}=\left(\mathbf{1,1},\mathbf{3}\right)_{4}\oplus\left(\mathbf{4,2,1}\right)_{3}\oplus\left(\mathbf{6,1},\mathbf{3}^{\prime}\right)_{2}\oplus\left(\mathbf{4}^{\prime}\mathbf{,2},\mathbf{3}\right)_{1}\,$.

A Jordan algebraic interpretation of the chains $1$ and $2$ is based on the following identification :

[TABLE]

and it reads as follows:

[TABLE]

[TABLE]

The second step of the Jordan-algebraic interpretation (5.161) of the chain $2$ is intentionally left explicit : intriguingly, it constitutes an enhancement of the symmetry obtained by the $D=5$ case of the Ehlers* *embedding for the $\left(\mathbb{R\oplus}J_{2}^{\mathbb{H}}\right)$-based theory. In fact, this latter reads

[TABLE]

and then the following enhancement takes place:

[TABLE]

The enhancement (5.163) actually hints for another interpretation of the non-symmetric and next-to-maximal embedding

[TABLE]

In fact, (5.164) can be interpreted as the $D=6$ case of the Ehlers embedding for the $\mathbb{H}$-based magic supergravity theory (which also enjoys a twin $\mathcal{N}=6$ fermionic completion), and its next-to-maximal nature is consistent with the treatment of [50]; in other words, (5.164) can be interpreted as follows:

[TABLE]

where therefore one of the $so(5,1)$ factor is conceived as $str_{0}\left(J_{2}^{\mathbb{H}}\right)$, while $so(3,3)$ is nothing but the $D=4$ Ehlers symmetry, and the commuting $su(2)$ factor is seen as $\mathcal{A}_{4}$. As a consequence of this interpretation, (5.161) can concisely be rewritten as :

[TABLE]

where

[TABLE]

5.7 $E_{7(-25)}$

This is the minimally non-compact real form of $E_{7}$. Its Jordan interpretation reads:

[TABLE]

5.7.1 ${\cal P}_{1}^{7(-25)}$

The maximal parabolics  ${\cal P}_{1}^{7(-25)}$   from (3.30) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

which gives rise to a $5$-grading with $\mathcal{N}_{1}^{7(-25)+}=\mathbf{1}_{2}\oplus\mathbf{32}_{1}^{\prime}\,$.

A Jordan algebraic interpretation (of the first step) of the chain above is given by:

[TABLE]

In this case,, the parabolic $so(1,1)$ (giving rise to the $5$-grading) is the non-compact Cartan of the $sl(2,\mathbb{R})$ factor in the first step of (5.170), namely of the axio-dilatonic ($S$-duality) factor of $conf\left(\mathbb{R}\oplus\mathfrak{J}_{2}^{\mathbb{O}}\right)$.

5.7.2 ${\cal P}_{2}^{7(-25)}$

The maximal parabolics  ${\cal P}_{2}^{7(-25)}$   from (3.30) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

Thus, it gives rise to a $5$-grading (cfr. (5.117)-(5.5.5), as well as (5.137)-(5.6.2)), with $\mathcal{N}_{2}^{7(-25)+}=(\mathbf{10},\mathbf{1})_{2}\oplus\left(\mathbf{16}^{\prime}\mathbf{,2}\right)_{1}\,$. Since

[TABLE]

a Jordan algebraic interpretation of the chain above is given by

[TABLE]

5.7.3 ${\cal P}_{3}^{7(-25)}$

The maximal parabolics  ${\cal P}_{3}^{7(-25)}$   from (3.30) corresponds to the Bruhat decomposition:

[TABLE]

giving rise to a $3$-grading with $\mathcal{N}_{3}^{7(-25)+}=\mathbf{27}_{2}^{\prime}\,$. A Jordan algebraic interpretation of the chain above reads

[TABLE]

therefore, the parabolic $so(1,1)$ (generating the $3$-grading) is the $so(1,1)_{KK}$ of the $S^{1}$-reduction $D=5\rightarrow 4$.

5.8 $E_{8(8)}$

This is the split real form of $E_{8}$. Its Jordan interpretation reads:

[TABLE]

5.8.1 ${\cal P}_{1}^{8(8)}$

The maximal parabolics  ${\cal P}_{1}^{8(8)}$   from (3.34) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through embedding chain:

[TABLE]

It gives rise to a $5$-grading, with ${\cal N}_{1}^{\,8(8)+}=\mathbf{14}_{2}\oplus\mathbf{64}_{1}^{\prime}\,$.   By observing that

[TABLE]

a Jordan algebraic interpretation (of the first step) of the chain above is given by the embedding $J_{3}^{\mathbb{O}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}$ considered at the $qconf$ level :

[TABLE]

5.8.2 ${\cal P}_{2}^{8(8)}$

The maximal parabolics  ${\cal P}_{2}^{8(8)}$   from (3.34) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

Thus, it gives rise to a $9$-grading, with  ${\cal N}_{2}^{\,8(8)+}=\left(\mathbf{7,1}\right)_{12}\oplus\left(\mathbf{7,2}\right)_{9}\oplus\left(\mathbf{35,1}\right)_{6}\oplus\left(\mathbf{21}^{\prime}\mathbf{,2}\right)_{3}\,$.  The interpretation of the first step of the chain above is provided by the $D=11$ case of the Ehlers embedding, which is actually relevant for $M$-theory (whose $g_{D=11}=\varnothing$) (cfr. [50], and Refs. therein) :

[TABLE]

5.8.3 ${\cal P}_{3}^{8(8)}$

The maximal parabolics  ${\cal P}_{3}^{8(8)}$   from (3.34) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained by at least four chains of embeddings171717Once again, we use the conventions of [91]., respectively denoted by $1$, $2$, $3$ and $4$ :

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The chains of embeddings $1$, $2$, $3$ and $4$ give rise to a $13$-grading, with

[TABLE]

all of real dimension $106$. Note that $1\leftrightarrow 3\equiv d)$ iff $\mathbf{3}\leftrightarrow\mathbf{3}^{\prime}$ in $sl(3,\mathbb{R})$, and $2\leftrightarrow 3\equiv 4$ iff the weights of the parabolic $so(1,1)$ of $2$ are multiplied by $1/2$, and iff $\mathbf{5}\leftrightarrow\mathbf{5}^{\prime}$ and $\mathbf{3}\leftrightarrow\mathbf{3}^{\prime}$ in $sl(5,\mathbb{R})$ and in $sl(3,\mathbb{R})$, respectively.

The Jordan algebraic interpretation of the chains $1$, $2$, $3$ and $4$ goes as follows. The first (non-symmetric) embedding in chain $1$ can be interpreted as the $D=7$ case of the Ehlers embedding for the $\mathbb{O}_{s}$-based theory (maximal supergravity) [50], thus determining the $D=7$ Ehlers $sl(5,\mathbb{R})_{Ehlers}$; then, the subsequent embedding enjoys a(n at least) twofold interpretation : it can be conceived as the uplift to $D=8$, where the $U$-duality Lie algebra is $g_{D=8}\left(\mathbb{O}_{s}\right)=sl(3,\mathbb{R})\oplus sl(2,\mathbb{R})$ (as such, the parabolic $so(1,1)$ in this chain is nothing but the KK $so(1,1)_{KK}$ in the $S^{1}$-reduction $D=8\rightarrow 7$ of maximal supergravity), or it can be seen as a further decomposition of $sl(5,\mathbb{R})_{Ehlers}$ into $sl(3,\mathbb{R})\oplus sl(2,\mathbb{R})\oplus so(1,1)$ :

[TABLE]

On the other hand, the first embedding in chain $2$ enjoys a(n at least) twofold interpretation, by virtue of the twofold characterization of $E_{7(7)}$ given by (5.68)-(5.69) (ultimately due to the symmetry of the double-split Magic Square $\mathcal{L}_{3}\left(\mathbb{A}_{s},\mathbb{B}_{s}\right)$ [37]) : it can be interpreted as the Ehlers embedding for the $\mathbb{O}_{s}$-based theory (corresponding to the uplift $D=3\rightarrow 4$ of maximal supergravity, and giving rise to the $D=4$ Ehlers symmetry $sl(2,\mathbb{R})_{Ehlers}$), or it can be conceived as a consequence of the embedding $\mathbb{O}_{s}\supset\mathbb{H}_{s}$, evaluated at the $qconf$ level. The second step also has an at least twofold interpretation, given by (5.128)-(5.129) :

[TABLE]

The first embedding in chain $3$ is the same as the first embedding of chain $2$. The second step consists of a maximal, non-symmetric embedding, and it has a(n at least) twofold interpretation, by virtue of the twofold characterization of $sl(6,\mathbb{R})$ given by (5.94) (once again due to the symmetry of $\mathcal{L}_{3}\left(\mathbb{A}_{s},\mathbb{B}_{s}\right)$ [37]) : it can be interpreted as the JP* *embedding for the $\mathbb{H}_{s}$-based theory (thus introducing the $D=5$ Ehlers $sl(3,\mathbb{R})_{Ehlers}$), or as the consequence of the (non-maximal) embedding $\mathbb{O}_{s}\supset\mathbb{C}_{s}$, evaluated at the level of conformal symmetries :

[TABLE]

Finally, each of the steps of the chain $4$ has an at least twofold interpretation. As a consequence of the twofold characterization of $E_{6(6)}$ given by (5.1)-(5.2) (once again, due to the symmetry of $\mathcal{L}_{3}\left(\mathbb{A}_{s},\mathbb{B}_{s}\right)$ [37]), a first line of interpretation (the upper one in (5.229)) conceives the first embedding as the JP embedding for the $\mathbb{O}_{s}$-based theory (corresponding to the uplift $D=3\rightarrow 5$ of maximal supergravity, and thus giving rise to the $D=5$ Ehlers symmetry $sl(3,\mathbb{R})_{Ehlers}$), followed by the consequence of the embedding $\mathbb{O}_{s}\supset\mathbb{H}_{s}$ at the level of reduced structure symmetry. A second line of interpretation (the lower one in (5.229)) sees the first embedding as a consequence of the non-maximal embedding $\mathbb{O}_{s}\supset\mathbb{C}_{s}$, evaluated at the $qconf$ level, and then followed an Ehlers embedding, corresponding to an uplift $D=3\rightarrow 4$, and therefore introducing the $D=4$ $sl(2,\mathbb{R})_{Ehlers}$ group. Note that the first two steps of chain $4$ realize the $D=8$ case of the Ehlers embedding for the $\mathbb{O}_{s}$-based theory [50]; thus, in this view the $sl(6,\mathbb{R})$ occurring in the second step is nothing but the $D=8$ Ehlers group :

[TABLE]

5.8.4 ${\cal P}_{4}^{8(8)}$

The maximal parabolics  ${\cal P}_{4}^{8(8)}$   from (3.34) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained by at least two chains of embeddings, respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

The chains of embeddings $1$ and $2$ give rise to an $11$-grading, with ${\cal N}_{4\,;\,a)}^{\,8(8)+}=(\mathbf{1},\mathbf{4})_{5}\oplus(\mathbf{10}^{\prime},\mathbf{1})_{4}\oplus(\mathbf{5},\mathbf{4}^{\prime})_{3}\oplus(\mathbf{5}^{\prime},\mathbf{6})_{2}\oplus(\mathbf{10},\mathbf{4})_{1}\,$,  ${\cal N}_{4\,;\,b)}^{\,8(8)+}=(\mathbf{1},\mathbf{4})_{15}\oplus(\mathbf{10}^{\prime},\mathbf{1})_{12}\oplus(\mathbf{5},\mathbf{4}^{\prime})_{9}\oplus(\mathbf{5}^{\prime},\mathbf{6})_{6}\oplus(\mathbf{10},\mathbf{4})_{3}$, both of real dimension $104$. Note that $1\leftrightarrow 2$ iff the weights of the parabolic $so(1,1)$ of $1$ are multiplied by $3$. The first steps of chains $1$ and $2$ are the same as the first steps of chains $1$ of subsubsection 5.8.3 and of subsubsection 5.8.2 above, respectively, and thus they correspondingly enjoy the same Jordan algebraic interpretation (in particular, (5.193) for the latter).

5.8.5 ${\cal P}_{5}^{8(8)}$

The maximal parabolics  ${\cal P}_{5}^{8(8)}$   from (3.34) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two embedding chains, respectively denoted by $1$ and $2$ (recall that $so(3,3)\cong sl(4,\mathbb{R})$):

[TABLE]

[TABLE]

The chains of embeddings $1$ and $2$ give rise to a $9$-grading, with ${\cal N}_{3\,;\,a)}^{\,8(8)+}=(\mathbf{1},\mathbf{3})_{4}\oplus(\mathbf{16}^{\prime},\mathbf{1})_{3}\oplus(\mathbf{10},\mathbf{3}^{\prime})_{2}\oplus(\mathbf{16},\mathbf{3})_{1}$,   ${\cal N}_{3\,;\,b)}^{\,8(8)+}=(\mathbf{1},\mathbf{3}^{\prime})_{4}\oplus(\mathbf{16},\mathbf{1})_{3}\oplus(\mathbf{10},\mathbf{3})_{2}\oplus(\mathbf{16}^{\prime},\mathbf{3}^{\prime})_{1}$, both of real dimension $97$. Note that $1\leftrightarrow 2$ iff the weights of the parabolic $so(1,1)$’s gets flipped (or, equivalently, iff all $\ {\cal M}_{5}^{8(8)}$-irreps. get conjugated). The Jordan-algebraic interpretation of chains $1$ and $2$ is based on the characterization (5.185) and on

[TABLE]

The first step of chain $1$ is a consequence of the embedding $J_{3}^{\mathbb{O}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}_{s}}$, considered at the $qconf$ level; then, by virtue of the quasi-conformal interpretation (5.249) of $so(5,5)$, the second step is a consequence (for cubic semi-simple Jordan algebras) of the non-maximal embedding $\mathbb{O}_{s}\supset\mathbb{C}_{s}$ (note the commuting factor $sl(4,\mathbb{R})$). Note also that the next-to-maximal (non-symmetric) embedding

[TABLE]

is the $D=6$ case of the Ehlers embedding [50] for the $\mathbb{O}_{s}$-based theory (maximal supergravity), thus characterizing $sl(4,\mathbb{R})$ as the $D=6$ Ehlers symmetry (cfr. (5.251) below). Concerning the chain $2$, it has ( at least) twofold interpretation : in the first interpretation, its first step is the JP embedding for the $\mathbb{O}_{s}$-based theory (maximal supergravity), thus giving rise to the $D=5$ Ehlers $sl(3,\mathbb{R})_{Ehlers}$; then, in the second step a further uplift to $D=6$ is performed, introducing the KK $so(1,1)_{KK}$ of the $S^{1}$-reduction $D=6\rightarrow 5$, which is the parabolic $so(1,1)$ in this chain. In the second interpretation, the first step is a consequence (for cubic simple Jordan algebras) of the non-maximal embedding $\mathbb{O}_{s}\supset\mathbb{C}_{s}$ (note the commuting factor $sl(3,\mathbb{R})$), while the second step is a consequence of the embedding $J_{3}^{\mathbb{C}_{s}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{C}_{s}}$, considered for quasi-conformal symmetries.

[TABLE]

[TABLE]

5.8.6 ${\cal P}_{6}^{8(8)}$

The maximal parabolics  ${\cal P}_{6}^{8(8)}$   from (3.34) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two chains of embeddings, respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

Both chains of embeddings $1$ and $2$ give rise to a $7$-grading, with  ${\cal N}_{3\,;\,a)}^{\,8(8)+}\cong{\cal N}_{3\,;\,b)}^{\,8(8)+}=(\mathbf{1},\mathbf{2})_{3}\oplus\left(\mathbf{27}^{\prime},\mathbf{1}\right)_{2}\oplus(\mathbf{27},\mathbf{2})_{1}$, with real dimension $83$. By recalling *e.g. *(5.217), the Jordan-algebraic interpretation of the chains $1$ and $2$ goes as follows :

[TABLE]

[TABLE]

5.8.7 ${\cal P}_{7}^{8(8)}$

The maximal parabolics  ${\cal P}_{7}^{8(8)}$   from (3.34) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

giving rise to a $3$-grading, with ${\cal N}_{7}^{\,8(8)+}=\mathbf{1}_{2}\oplus\mathbf{56}_{1}$, with real dimension $57$. The Jordan algebraic interpretation of the first step is the same as the chain $1$ of case 5.8.6 above.

5.8.8 ${\cal P}_{8}^{8(8)}$

The maximal parabolics  ${\cal P}_{8}^{8(8)}$   from (3.34) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

giving rise to a $7$-grading, with ${\cal N}_{8}^{\,8(8)+}=\mathbf{8}_{9}\oplus\mathbf{28}_{6}^{\prime}\oplus\mathbf{56}_{3}$, with real dimension $92$.

The Jordan algebraic interpretation of the chain goes as follows (recall (5.193))

[TABLE]

5.9 $E_{8(-24)}$

This is the minimally non-compact real form of $E_{8}$. Its Jordan interpretation reads:

[TABLE]

5.9.1 ${\cal P}_{1}^{8(-24)}$

The maximal parabolics  ${\cal P}_{1}^{8(-24)}$   from (3.43) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

giving rise to a $5$-grading, with ${\cal N}_{8}^{\,8(-24)+}=\mathbf{14}_{2}\oplus\mathbf{64}_{1}^{\prime}$. Since

[TABLE]

a Jordan algebraic interpretation (of the first step) of the chain above is given by the embedding $J_{3}^{\mathbb{O}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}}$ considered at the $qconf$ level :

[TABLE]

5.9.2 ${\cal P}_{2}^{8(-24)}$

The maximal parabolics  ${\cal P}_{2}^{8(-24)}$   from (3.43) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two chains of embeddings, respectively denoted by $1$ and $2$ (recall that $so(3,3)\cong sl(4,\mathbb{R})$) :

[TABLE]

[TABLE]

Both chains $1$ and $2$ give rise to a $9$-grading, with  ${\cal N}_{2\,;\,a)}^{\,8(-24)+}=(\mathbf{1},\mathbf{3})_{4}\oplus(\mathbf{16}^{\prime},\mathbf{1})_{3}\oplus(\mathbf{10},\mathbf{3}^{\prime})_{2}\oplus(\mathbf{16},\mathbf{3})_{1}$,  ${\cal N}_{2\,;\,b)}^{\,8(-24)+}=(\mathbf{1},\mathbf{3}^{\prime})_{4}\oplus(\mathbf{16},\mathbf{1})_{3}\oplus(\mathbf{10},\mathbf{3})_{2}\oplus(\mathbf{16}^{\prime},\mathbf{3}^{\prime})_{1}$, both of real dimension $97$. Note that $1\leftrightarrow 2$ iff the weights of the parabolic $so(1,1)$’s gets flipped (or, equivalently, iff all $\mathcal{M}_{5}^{8(8)}$-irreps. get conjugated). The Jordan-algebraic interpretation of chains $1$ and $2$ is based on the identification (5.270) and on

[TABLE]

Note that $so(9,1)$, differently from the split form $so(5,5)$ (cfr. (5.249)), does not admit a quasi-conformal interpretation. The first step of chain $1$ is then a consequence of the embedding $J_{3}^{\mathbb{O}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{O}}$, considered at the $qconf$ level. Note that the next-to-maximal (non-symmetric) embedding

[TABLE]

is the $D=6$ case of the Ehlers embedding [50] for the $\mathbb{O}$-based theory, thus characterizing $sl(4,\mathbb{R})$ as the $D=6$ Ehlers symmetry (cfr. (5.279) below). Concerning the chain $2$, its first step is the JP embedding for the $\mathbb{O}$-based theory, thus giving rise to the $D=5$ Ehlers $sl(3,\mathbb{R})_{Ehlers}$; then, in the second step a further uplift to $D=6$ is performed, introducing the KK $so(1,1)_{KK}$ of the $S^{1}$-reduction $D=6\rightarrow 5$, which is the parabolic $so(1,1)$ in this chain:

[TABLE]

[TABLE]

5.9.3 ${\cal P}_{3}^{8(-24)}$

The maximal parabolics  ${\cal P}_{3}^{8(-24)}$   from (3.43) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two chains of embeddings, respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

Both chains $1$ and $2$ give rise to a $7$-grading, with ${\cal N}_{2\,;\,1}^{\,8(-24)+}\cong{\cal N}_{2\,;\,1}^{\,8(-24)+}=(\mathbf{1},\mathbf{2})_{3}\oplus(\mathbf{27}^{\prime},\mathbf{1})_{2}\oplus(\mathbf{27}^{\prime},\mathbf{2})_{1}$, with real dimension $83$.

The Jordan algebraic interpretation of the chains $1$ and $2$ goes as follows :

[TABLE]

[TABLE]

5.9.4 ${\cal P}_{4}^{8(-24)}$

The maximal parabolics  ${\cal P}_{4}^{8(-24)}$   from (3.43) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

giving rise to a $5$-grading, with ${\cal N}_{4}^{\,8(-24)+}=\mathbf{1}_{2}\oplus\mathbf{56}_{1}$, with real dimension $57$.

The Jordan algebraic interpretation of the first step is the same as the chain $1$ of subsubsection 5.9.3, and it is analogous to the chain from the parabolically related case  ${\cal P}_{7}^{8(8)}\,$, Subsubsection 5.8.7.

5.10 $F_{4(4)}$

This is the split real form of $F_{4}$. Its Jordan interpretation is twofold (due to the symmetry of the double-split Magic Square ${\mathcal{L}}_{3}(\mathbb{A}_{s},\mathbb{B}_{s})$ [37], reported in Table 3) :

[TABLE]

5.10.1 ${\cal P}_{1}^{4(4)}$

The maximal parabolics  ${\cal P}_{1}^{4(4)}$   from (3.48) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two chains of embeddings respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

[TABLE]

Both chains $1$ and $2$ give rise to a $9$-grading, with  ${\cal M}_{1}^{4(4)}~{}\cong~{}sl(3,\mathbb{R})_{S}\oplus sl(2,\mathbb{R})_{L}\,$ and  ${\cal N}_{1\,;\,1}^{\,4(4)+}\cong{\cal N}_{1\,;\,2}^{\,4(4)+}=(\mathbf{3},\mathbf{1})_{4}\oplus(\mathbf{1},\mathbf{2})_{3}\oplus(\mathbf{3}^{\prime},\mathbf{3})_{2}\oplus(\mathbf{3},\mathbf{2})_{1}$, of real dimension $20$.

Since

[TABLE]

a Jordan algebraic interpretation (of the first step) of the chain $1$ is given by the embedding $J_{3}^{\mathbb{R}}\supset\mathbb{R}\oplus J_{2}^{\mathbb{R}}$ considered at the $qconf$ level :

[TABLE]

The second step realizes (with respect to $F_{4(4)}$) the $D=6$ case of the Ehlers embedding for the $\mathbb{R}$-based theory, thus giving rise to the $D=6$ Ehlers symmetry $sl(4,\mathbb{R})_{Ehlers}$ (which then gets further branched in order to generate the parabolic $so(1,1)$).

Finally, the first step of the chain $2$ can be interpreted as the JP embedding for the $\mathbb{R}$-based theory, where $sl(3,\mathbb{R})_{L}\cong str_{0}\left(J_{3}^{\mathbb{R}}\right)$, and $sl(3,\mathbb{R})_{S}\cong sl(3,\mathbb{R})_{Ehlers}$ is the $D=5$ Ehlers symmetry. Furthermore, $sl(3,\mathbb{R})_{L}\cong str_{0}\left(J_{3}^{\mathbb{R}}\right)$ branches into $sl(2,\mathbb{R})_{L}\cong so(2,1)\cong str_{0}\left(J_{2}^{\mathbb{R}}\right)\cong str_{0}\left(\mathbb{R}\oplus J_{2}^{R}\right)\ominus so(1,1)$, thus allowing for the identification of the parabolic $so(1,1)$ with the $so(1,1)_{KK}$ of the $D=6\rightarrow 5$ $S^{1}$-reduction.

5.10.2 ${\cal P}_{2}^{4(4)}$

The maximal parabolics  ${\cal P}_{2}^{4(4)}$   from (3.48) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two chains of embeddings, respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

[TABLE]

Both chains of embeddings $1$ and $2$ give rise to a $7$-grading, with

 ${\cal M}_{2}^{4(4)}~{}\cong~{}sl(3,\mathbb{R})_{L}\oplus sl(2,\mathbb{R})_{S}\,$ and  ${\cal N}_{2\,;\,b)}^{\,4(4)+}\cong{\cal N}_{2\,;\,c.1)}^{\,4(4)+}=(\mathbf{1},\mathbf{2})_{3}\oplus(\mathbf{6},\mathbf{1})_{2}\oplus(\mathbf{6}^{\prime},\mathbf{2})_{1}$, of real dimension $20$.

Since

[TABLE]

the Jordan algebraic interpretation of the chain $1$ reads:

[TABLE]

where $sl(2,\mathbb{R})_{Ehlers}$ is the $D=4$ Ehlers symmetry.

Finally, the first step of the chain $2$ can be interpreted as the JP embedding for the $\mathbb{R}$-based theory, where $sl(3,\mathbb{R})_{L}\cong str_{0}\left(J_{3}^{\mathbb{R}}\right)$, and $sl(3,\mathbb{R})_{S}\cong sl(3,\mathbb{R})_{Ehlers}$ is the $D=5$ Ehlers symmetry. (This is the same as in Subsubsection 5.10.1.) Then, $sl(3,\mathbb{R})_{S}=sl(3,\mathbb{R})_{Ehlers}$ branches to  $sl(2,\mathbb{R})_{S}$  to generate the parabolic $so(1,1)$.

5.10.3 ${\cal P}_{3}^{4(4)}$

The maximal parabolics  ${\cal P}_{3}^{4(4)}$   from (3.48) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

giving rise to a $5$-grading, with   ${\cal N}_{3}^{\,4(4)+}=\mathbf{1}_{2}\oplus\mathbf{14}_{1}^{\prime}$, with real dimension $15$.

The Jordan algebraic interpretation of the first step is the same as the chain $1$ of subsubsection 5.10.2 above. Thus, the parabolic $so(1,1)$ can here be interpreted as the non-compact Cartan generator of the $D=4$ Ehlers symmetry $sl(2,\mathbb{R})_{Ehlers}$.

5.10.4 ${\cal P}_{4}^{4(4)}$

The maximal parabolics  ${\cal P}_{4}^{4(4)}$   from (3.48) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through the embedding chain

[TABLE]

giving rise to a $5$-grading, with  ${\cal N}_{4}^{\,4(4)+}=\mathbf{7}_{2}\oplus\mathbf{8}_{1}$, with real dimension $15$.

The Jordan algebraic interpretation of the first step is the same as the chain $1$ of subsubsection 5.10.1 above. Moreover, by observing that

[TABLE]

the second step can be interpreted as the consequence, at the $qconf$ level, of the embedding $\mathbb{R}\oplus J_{2}^{\mathbb{R}}\supset$ $\mathbb{R}\oplus\mathbb{R}$, corresponding to the embedding of the $c$-map [49] of the so-called $ST^{2}$ model of $\mathcal{N}=2$, $D=4$ supergravity into the $c$-map of the $\left(\mathbb{R}\oplus J_{2}^{\mathbb{R}}\cong\mathbb{R}\oplus\Gamma_{1,2}\right)$-based model of $\mathcal{N}=2$, $D=4$ supergravity.

5.11 $F_{4(-20)}$

The Jordan interpretation of $F_{4(-20)}$ reads:

[TABLE]

where $J_{2,1}^{\mathbb{O}}\cong J_{1,2}^{\mathbb{O}}$ denotes the rank-$3$ Lorentzian Jordan algebra over $\mathbb{O}$ (cfr. e.g. [92, 36], and Refs. therein); indeed, the non-Euclidean nature of $J_{2,1}^{\mathbb{O}}$ generally implies the non-compactness of its automorphism group (differently from the automorphism symmetry of Euclidean Jordan algebras over division algebras, which is always compact). Moreover, $\mathcal{S}\left(J_{3}^{\mathbb{O}}\right)$ denotes the stabilizer group of the rank-$3$ orbit of the action of $Str_{0}\left(J_{3}^{\mathbb{O}}\right)$ on its (fundamental) irrep. $\mathbf{27}$ with non-vanishing cubic invariant $I_{3}\neq 0$ and representative “$++-$” (for further detail, cfr. e.g. [93, 88]; the relation between $\mathcal{S}$ and $\mathcal{K}$ is investigated in [87]).

The maximal parabolics  ${\cal P}_{2}^{4(-20)}$   from (3.57) corresponds to the Bruhat decomposition (which is both maximal and minimal):

[TABLE]

which can be obtained through the embedding chain

[TABLE]

giving rise to a $5$-grading, with ${\cal N}^{\,4(-20)+}=\mathbf{7}_{2}\oplus\mathbf{8}_{1}$, with real dimension $15$. Note that $so(8,1)$ does not have a quasi-conformal interpretation. However, we observe that

[TABLE]

which is the $q=8$ case of the general result

[TABLE]

Therefore, the first step of the chain can be interpreted as the consequence, at the level of the $\mathcal{S}$-symmetry, of the embedding $J_{3}^{\mathbb{R}}\supset$ $\mathbb{R}\oplus J_{2}^{\mathbb{R}}$ :

[TABLE]

5.12 $G_{2(2)}$

The Jordan interpretation of $G_{2(2)}$, split real form of $G_{2}$, reads:

[TABLE]

where $\mathbb{R}$ here denotes the real numbers, conceived as the simplest example of cubic simple Jordan algebra with cubic norm (the parameter $q$ for this case has the effective value $q=-2/3$; see e.g. [94, 95], and Refs. therein).

The maximal parabolics  ${\cal P}_{LS}^{2(2)}$   from (3.59) corresponds to the Bruhat decomposition:

[TABLE]

which can be obtained through at least two chains of embeddings, respectively denoted by $1$ and $2$ :

[TABLE]

[TABLE]

The chains of embeddings $1$ and $2.2$ give rise to a $7$-grading, with  ${\cal M}_{S}^{2(2)}=sl(2,\mathbb{R})_{S}$  and  ${\cal N}_{1)}^{\,2(2)+}\cong{\cal N}_{2.2)}^{\,2(2)+}\cong{\cal N}_{S}^{\,2(2)+}=\mathbf{2}_{3}\oplus\mathbf{1}_{2}\oplus\mathbf{2}_{1}\,$, with real dimension $5$. On the other hand, the chain of embeddings $2.1$ gives rise to a $5$-grading, with  ${\cal M}_{L}^{2(2)}=sl(2,\mathbb{R})_{L}$  and  ${\cal N}_{2.1)}^{\,2(2)+}\cong{\cal N}_{L}^{\,2(2)+}=\mathbf{1}_{2}\oplus\mathbf{4}_{1}\,$, once again with real dimension $5$.

The Jordan algebraic interpretation of the first step of chain $1$ is provided by the JP embedding for the $\mathbb{R}$-based theory under consideration, recalling that $str_{0}\left(\mathbb{R}\right)\cong\varnothing$. On the other hand, by observing that

[TABLE]

the first step of chain $2$ can be conceived as the Ehlers embedding for the $\mathbb{R}$-based theory under consideration, followed by two possible second steps : in $2.1$ the $D=4$ Ehlers group $sl(2,\mathbb{R})_{L}\cong sl(2,\mathbb{R})_{Ehlers}$ is branched, and thus the parabolic $so(1,1)$ can be interpreted as its non-compact Cartan; in $2.2$ $sl(2,\mathbb{R})_{S}\cong conf\left(\mathbb{R}\right)$ gets branched, and therefore the parabolic $so(1,1)$ is the KK $so(1,1)$ of the $S^{1}$-reduction $D=5\rightarrow 4$ for the theory at stake (which is the so-called $T^{3}$ model of $\mathcal{N}=2$, $D=4$ supergravity [98]).

[TABLE]

[TABLE]

6 Outlook

In the present paper we initiated the investigation of the relations between representation theory and Jordan algebras, focussing on non-compact real forms of finite-dimensional exceptional Lie algebras. We provided a derivation of the maximal parabolic subalgebras in terms of chains of maximal (symmetric or non-symmetric) embeddings of Lie algebras, which were then interpreted in terms of symmetries of Jordan algebras (in particular, we focussed on the rank-2 and rank-3 classes). This also allowed to provide a complete Jordan algebraic characterization (classified in Table B) of the maximally parabolical relations between exceptional Lie algebras (classified in Table A).

There is a number of possible venues for further future research. For instance, in light of the Jordan algebraic interpretation provided in this paper, the relevance of maximal parabolic subalgebras to the theory of induced representations might have interesting consequences in Maxwell-Einstein (super)gravity $\mathcal{N}$-extended theories in various space-time dimensions. Moreover, the present analysis might be extended to classical Lie algebras (and to higher-rank Jordan algbras), and slight generalizations of the definition of maximal parabolic subalgebras are possible within the Borel-de Siebenthal theory. We plan to deal with such issues in future works [99].

Acknowledgments

V.K.D. acknowledges partial support from Bulgarian NSF Grant DN-18/1.