Quantum Deformations of Relativistic Symmetries

TL;DR

This paper explores quantum deformations of relativistic symmetries, specifically Lorentz and Poincare algebras, identifying how classical r-matrices relate to twisted deformations and providing explicit examples.

Contribution

It classifies quantum deformations of D=4 Lorentz and Poincare algebras, linking Zakrzewski's r-matrices to specific twisted deformation types and providing explicit twist constructions.

Findings

Most Zakrzewski r-matrices correspond to Abelian and Jordanian twists.

Explicit forms of some twists are provided.

The work enhances understanding of quantum symmetry deformations.

Abstract

We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.

Equations

$$[r^{12},r^{13}+r^{23}]+[r^{13},r^{23}]$$

$$r_A$$

$$F_{r_A}$$

$$F^{12}(\Delta \otimes \mathrm{id})(F)=F^{23}(\mathrm{id}\otimes \Delta )(F),$$

$$(ϵ\otimes \mathrm{id})(F)=(\mathrm{id}\otimes ϵ)(F)=1.$$

$${\Delta }^{(F)}(a)=F\Delta (a)F^{-1},S^{(F)}(a)=uS(a)u^{-1}$$

$$r_{J}(\xi)\;=\;\xi\,\Bigl{(}\sum_{\nu=0}^{n}x_{\nu}\wedge y_{\nu}\Bigr{)}~{},$$

$$\begin{array}[]{rcl}[x_{0},\,y_{0}]\!\!&=\!\!&y_{0}~{},\qquad[x_{0},\,x_{i}]\;=\;(1-t_{i})x_{i}~{},\qquad\;\;[x_{0},\,y_{i}]\;=\;t_{i}y_{i}~{},\\[7.0pt] [x_{i},\,y_{j}]\!\!&=\!\!&\delta_{ij}y_{0}~{},\quad[x_{i},\,x_{j}]\;=\;[y_{i},\,y_{j}]\;=\;0~{},\quad\;[y_{0},\,x_{j}]\;=\;[y_{0},\,y_{j}]\;=\;0~{},\end{array}$$

$$F_{r_J}$$

$${\delta }_{r_1}(x):=[x\otimes 1+1\otimes x,r_1]=0,\forall x\in \mathrm{Sup}(r_2).$$

$$F_r$$

$$F_r(\xi )$$

$$[h,e_{\pm }]=\pm e_{\pm },[e_{+},e_{-}]=2h,$$

$$[h,e_{\pm }^{\prime }]=\pm e_{\pm }^{\prime },[h^{\prime },e_{\pm }]=\pm e_{\pm }^{\prime },[e_{\pm },e_{\mp }^{\prime }]=\pm 2h^{\prime },$$

$$[h^{\prime },e_{\pm }^{\prime }]=\mp e_{\pm },[e_{+}^{\prime },e_{-}^{\prime }]=-2h,$$

$$x^{\ast }=-x(\forall x\in \mathfrak{o}(3,1)).$$

$$r_1$$

$$r_2$$

$$r_3$$

$$r_4$$

$$r_j^{\ast }=-r_j(j=1,2,3,4).$$

$$(x\otimes y{)}^{\ast }$$

$$F_{r_1}=\exp (h\otimes \sigma ),\sigma =\frac{1}{2}\ln (1+\alpha e_{+})$$

$$F_{r_{2}}\,=\,\exp{\Bigl{(}\frac{\imath\beta}{\alpha^{2}}\;\sigma\wedge\varphi\Bigr{)}}\,\exp{(h\otimes\sigma-h^{\prime}\otimes\varphi)}~{},$$

$$\sigma =\frac{1}{2}\ln \left[(1+\alpha e_{+}{)}^2+(\alpha e_{+}^{\prime }{)}^2\right],\phi =\arctan \frac{\alpha e_{+}^{\prime }}{1+\alpha e_{+}}$$

$$F_{r_{1}}^{\prime}=\,\exp{\Bigl{(}\frac{1}{2}\Delta(h)-\frac{1}{2}\Bigl{(}h\frac{\sinh\alpha e_{+}}{\alpha e_{+}}\otimes e^{-\alpha e_{+}}\!+e^{\alpha e_{+}}\!\otimes h\frac{\sinh\alpha e_{+}}{\alpha e_{+}}\Bigr{)}\frac{\alpha\Delta(e_{+})}{\sinh\alpha\Delta(e_{+})}\,\Bigr{)}},$$

$$H_1$$

$$H_2$$

$$[H_k,E_{k\pm }]=\pm E_{k\pm },[E_{k+},E_{k-}]=2H_k(k=1,2).$$

$$H_1^{\ast }=-H_2,E_{1\pm }^{\ast }=-E_{2\pm },H_2^{\ast }=-H_1,E_{2\pm }^{\ast }=-E_{1\pm }.$$

$$\begin{array}[]{rcl}r_{3}\!\!&=\!\!&r^{\prime}_{1}+r^{\prime\prime}_{1}~{},\\[7.0pt] r^{\prime}_{3}\!\!&:=\!\!&2(\beta+\imath\alpha)E_{1+}\wedge E_{1-}+2(\beta-\imath\alpha)E_{2+}\wedge E_{2-}~{},\\[7.0pt] r^{\prime\prime}_{3}\!\!&:=\!\!&4\imath\gamma\,H_{2}\wedge H_{1}~{},\end{array}$$

$$\begin{array}[]{rcl}r_{4}\!\!&=\!\!&r^{\prime}_{4}+r^{\prime\prime}_{4}~{},\\[7.0pt] r^{\prime}_{4}\!\!&:=\!\!&2\imath\alpha(E_{1+}\wedge E_{1-}-E_{2+}\wedge E_{2-}-2H_{1}\wedge H_{2})~{},\\[7.0pt] r^{\prime\prime}_{4}\!\!&:=\!\!&4\imath\lambda\,E_{1+}\wedge E_{2+}\end{array}$$

$$q_{z_{+}}^{H_1}{E}_{1\pm }$$

Full text

1 Introduction

The quantum deformations of relativistic symmetries are described by Hopf-algebraic deformations of Lorentz and Poincaré algebras. Such quantum deformations are classified by Lorentz and Poincaré Poisson structures. These Poisson structures given by classical $r$-matrices were classified already some time ago by S. Zakrzewski in [1] for the Lorentz algebra and in [2] for the Poincaré algebra. In the case of the Lorentz algebra a complete list of classical $r$-matrices involves the four independent formulas and the corresponding quantum deformations in different forms were already discussed in literature (see [3, 4, 5, 6, 7]). In the case of Poincaré algebra the total list of the classical $r$-matrices, which satisfy the homogeneous classical Yang-Baxter equation, consists of 20 cases which have various numbers of free parameters. Analysis of these twenty solutions shows that each of them can be presented as a sum of subordinated $r$-matrices which almost all are of Abelian and Jordanian types. A part of twists corresponding to the $r$-matrices of Zakrzewski classification are given in explicit form.

2 Preliminaries

Let $r$ be a classical $r$-matrix of a Lie algebra $\mathfrak{g}$, i.e. $r\in\,\stackrel{{\scriptstyle 2}}{{\wedge}}\mathfrak{g}$ and $r$ satisfies to the classical Yang–Baxter equation (CYBE)

[TABLE]

where $\Omega$ is $\mathfrak{g}$-invariant element, $\Omega\in(\stackrel{{\scriptstyle 3}}{{\wedge}}\mathfrak{g})_{\mathfrak{g}}$. We consider two types of the classical $r$-matrices and corresponding twists.

Let the classical $r$-matrix $r=r_{A}$ has the form

[TABLE]

where all elements $x_{i},y_{i}$ $(i=1,\ldots,n)$ commute among themselves. Such an $r$-matrix is called of Abelian type. The corresponding twist is given as follows

[TABLE]

This twisting two-tensor $F:=F_{r_{A}}$ satisfies the cocycle equation

[TABLE]

and the ”unital” normalization condition

[TABLE]

The twisting element $F$ defines a deformation of the universal enveloping algebra $U(\mathfrak{g})$ considered as a Hopf algebra. The new deformed coproduct and antipode are given as follows

[TABLE]

for any $a\in U(\mathfrak{g})$, where $\Delta(a)$ is a co-product before twisting, and $u=\sum_{i}f^{(1)}_{i}S(f^{(2)}_{i})$ if $F=\sum_{i}f^{(1)}_{i}\otimes f^{(2)}_{i}$.

Let the classical $r$-matrix $r=r_{J}(\xi)$ has the form333Here entering the parameter deformation $\xi$ is a matter of convenience.

[TABLE]

where the elements $x_{\nu},y_{\nu}$ $(\nu=0,1,\ldots,n)$ satisfy the relations444It is easy to verify that the two-tensor (2.7) indeed satisfies the homogenous classical Yang-Baxter equation (2.1) (with $\Omega=0$), if the elements $x_{\nu},y_{\nu}$ $(\nu=0,1,\ldots,n)$ are subject to the relations (2.8).

[TABLE]

$(i,j=1,\ldots,n)$, $(t_{i}\in{\mathbb{C}})$. Such an $r$-matrix is called of Jordanian type. The corresponding twist is given as follows [8, 9]

[TABLE]

where $\sigma:=\frac{1}{2}\ln(1+\xi y_{0})$.555The corresponding twists for Lie algebras $\mathfrak{sl}(n)$, $\mathfrak{so}(n)$ and $\mathfrak{sp}(2n)$ were firstly constructed in the papers [10, 11, 12, 13].

Let $r$ be an arbitrary $r$-matrix of $\mathfrak{g}$. We denote a support of $r$ by $\mathop{\rm Sup}(r)$666The support $\mathop{\rm Sup}(r)$ is a subalgebra of $\mathfrak{g}$ generated by the elements $\{x_{i},y_{i}\}$ if $r=\sum_{i}x_{i}\wedge y_{i}$.. The following definition is useful.

Definition 2.1

Let $r_{1}$ and $r_{2}$ be two arbitrary classical $r$-matrices. We say that $r_{2}$ is subordinated to $r_{1}$, $r_{1}\succ r_{2}$, if $\delta_{r_{1}}(\mathop{\rm Sup}(r_{2}))=0$, i.e.

[TABLE]

If $r_{1}\succ r_{2}$ then $r=r_{1}+r_{2}$ is also a classical $r$-matrix (see [15]). The subordination enables us to construct a correct sequence of quantizations. For instance, if the $r$-matrix of Jordanian type (2.7) is subordinated to the $r$-matrix of Abelian type (2.2), $r_{\!A}\succ r_{\!J}$, then the total twist corresponding to the resulting $r$-matrix $r=r_{\!A}+r_{\!J}$ is given as follows

[TABLE]

The further definition is also useful.

Definition 2.2

A twisting two-tensor $F_{r}(\xi)$ of a Hopf algebra, satisfying the conditions (2.4) and (2.5), is called locally $r$-symmetric if the expansion of $F_{r}(\xi)$ in powers of the parameter deformation $\xi$ has the form

[TABLE]

where $r$ is a classical $r$-matrix, and $c$ is a numerical coefficient, $c\neq 0$.

It is evident that the Abelian twist (2.3) is globally $r$-symmetric and the twist of Jordanian type (2.9) does not satisfy the relation (2.12), i.e. it is not locally $r$-symmetric.

3 Quantum deformations of Lorentz algebra

The results of this section in different forms were already discussed in literature (see [3, 4, 5, 6, 7]).

The classical canonical basis of the $D=4$ Lorentz algebra, $\mathfrak{o}(3,1)$, can be described by anti-Hermitian six generators ($h$, $e_{\pm}$, $h^{\prime}$, $e^{\prime}_{\pm}$) satisfying the following non-vanishing commutation relations777Since the real Lie algebra $\mathfrak{o}(3,1)$ is standard realification of the complex Lie $\mathfrak{sl}(2,\mathbb{C})$ these relations are easy obtained from the defining relations for $\mathfrak{sl}(2,\mathbb{C})$, i.e. from (3.1).:

[TABLE]

and moreover

[TABLE]

A complete list of classical $r$-matrices which describe all Poisson structures and generate quantum deformations for $\mathfrak{o}(3,1)$ involve the four independent formulas [1]:

[TABLE]

If the universal $R$-matrices of the quantum deformations corresponding to the classical $r$-matrices (3.5)–(3.8) are unitary then these $r$-matrices are anti-Hermitian, i.e.

[TABLE]

Therefore the $*$-operation (3.4) should be lifted to the tensor product $\mathfrak{o}(3,1)\otimes\mathfrak{o}(3,1)$. There are two variants of this lifting: direct and flipped, namely,

[TABLE]

We see that if the ”direct” lifting of the $*$-operation (3.4) is used then all parameters in (3.5)–(3.8) are pure imaginary. In the case of the ”flipped” lifting (3.11) all parameters in (3.5)–(3.8) are real.

The first two $r$-matrices (3.5) and (3.6) satisfy the homogeneous CYBE and they are of Jordanian type. If we assume (3.10), the corresponding quantum deformations were described detailed in the paper [6] and they are entire defined by the twist of Jordanian type:

[TABLE]

for the $r$-matrix (3.5), and

[TABLE]

[TABLE]

for the $r$-matrix (3.6). It should be recalled that the twists (3.12) and (3.13) are not locally $r$-symmetric. A locally $r$-symmetric twist for the $r$-matrix (3.5) was obtained in [14] and it has the following complicated formula:

[TABLE]

where $\Delta$ is a primitive coproduct.

The last two $r$-matrices (3.7) and (3.8) satisfy the non-homogeneous (modified) CYBE and they can be easily obtained from the solutions of the complex algebra $\mathfrak{o}(4,\mathbb{C})\simeq\mathfrak{sl}(2,\mathbb{C})\oplus\mathfrak{sl}(2,\mathbb{C})$ which describes the complexification of $\mathfrak{o}(3,1)$. Indeed, let us introduce the complex basis of Lorentz algebra $(\mathfrak{o}(3,1)\simeq\mathfrak{sl}(2;\mathbb{C})\oplus\mathfrak{\overline{sl}}(2,\mathbb{C}))$ described by two commuting sets of complex generators:

[TABLE]

which satisfy the relations (compare with (3.1))

[TABLE]

The $*$-operation describing the real structure acts on the generators $H_{k}$, and $E_{k\pm}$ ($k=1,2$) as follows

[TABLE]

The classical $r$-matrix $r_{3}$, (3.7), and $r_{4}$, (3.8), in terms of the complex basis (3.16), (3.17) take the form

[TABLE]

and

[TABLE]

For the sake of convenience we introduce parameter888We can reduce this parameter $\lambda$ to $\pm\frac{1}{2}$ by automorphism of $\mathfrak{o}(4,\mathbb{C})$.$\lambda$ in $r^{\prime\prime}_{4}$. It should be noted that $r^{\prime}_{3}$, $r^{\prime\prime}_{3}$ and $r^{\prime}_{4}$, $r^{\prime\prime}_{4}$ are themselves classical $r$-matrices. We see that the $r$-matrix $r^{\prime}_{3}$ is simply a sum of two standard $r$-matrices of $\mathfrak{sl}(2;\mathbb{C})$, satisfying the anti-Hermitian condition $r^{*}=-r$. Analogously, it is not hard to see that the $r$-matrix $r_{4}$ corresponds to a Belavin-Drinfeld triple [15] for the Lie algebra $\mathfrak{sl}(2;\mathbb{C})\oplus\mathfrak{\overline{sl}}(2,\mathbb{C}))$. Indeed, applying the Cartan automorphism $E_{2\pm}\rightarrow E_{2\mp}$, $H_{2}\rightarrow-H_{2}$ we see that this is really correct (see also [16]).

We firstly describe quantum deformation corresponding to the classical $r$-matrix $r_{3}$ (3.20). Since the $r$-matrix $r^{\prime\prime}_{3}$ is Abelian and it is subordinated to $r^{\prime}_{3}$ therefore the algebra $\mathfrak{o}(3,1)$ is firstly quantized in the direction $r^{\prime}_{3}$ and then an Abelian twist corresponding to the $r$-matrix $r^{\prime\prime}_{3}$ is applied. We introduce the complex notations $z_{\pm}:=\beta\pm\imath\alpha$. It should be noted that $z_{-}=z_{+}^{*}$ if the parameters $\alpha$ and $\beta$ are real, and $z_{-}=-z_{+}^{*}$ if the parameters $\alpha$ and $\beta$ are pure imaginary. From structure of the classical $r$-matrix $r_{3}^{\prime}$ it follows that a quantum deformation $U_{r^{\prime}_{1}}(\mathfrak{o}(3,1))$ is a combination of two $q$-analogs of $U(\mathfrak{sl}(2;\mathbb{C}))$ with the parameter $q_{z_{+}}$ and $q_{z_{-}}$, where $q_{z_{\pm}}:=\exp{z_{\pm}}$. Thus $U_{r^{\prime}_{3}}(\mathfrak{o}(3,1))\cong U_{q_{z_{+}}}(\mathfrak{sl}(2;\mathbb{C}))\otimes U_{q_{z_{-}}}(\overline{\mathfrak{sl}}(2;\mathbb{C}))$ and the standard generators $q_{z_{+}}^{\pm H_{1}}$, $E_{1\pm}$ and $q_{z_{-}}^{\pm H_{2}}$, $E_{2\pm}$ satisfy the following non-vanishing defining relations

[TABLE]

In this case the co-product $\Delta_{r^{\prime}_{1}}$ and antipode $S_{r^{\prime}_{1}}$ for the generators $q_{z_{+}}^{\pm H_{1}}$, $E_{1\pm}$ and $q_{z_{-}}^{\pm H_{2}}$, $E_{2\pm}$ can be given by the formulas:

[TABLE]

[TABLE]

The $*$-involution describing the real structure on the generators (3.8) can be adapted to the quantum generators as follows

[TABLE]

and there exit two $*$-liftings: direct and flipped, namely,

[TABLE]

for any $a\otimes b\in U_{r^{\prime}_{3}}(\mathfrak{o}(3,1))\otimes U_{r^{\prime}_{3}}(\mathfrak{o}(3,1))$, where the $*$-direct involution corresponds to the case of the pure imaginary parameters $\alpha,\,\beta$ and the $*$-flipped involution corresponds to the case of the real deformation parameters $\alpha,\,\beta$. It should be stressed that the Hopf structure on $U_{r^{\prime}_{3}}(\mathfrak{o}(3,1))$ satisfy the consistency conditions under the $*$-involution

[TABLE]

Now we consider deformation of the quantum algebra $U_{r^{\prime}_{3}}(\mathfrak{o}(3,1))$ (secondary quantization of $U(\mathfrak{o}(3,1))$) corresponding to the additional $r$-matrix $r^{\prime\prime}_{3}$, (3.20). Since the generators $H_{1}$ and $H_{2}$ have the trivial coproduct

[TABLE]

therefore the unitary two-tensor

[TABLE]

satisfies the cocycle condition (2.4) and the ”unital” normalization condition (2.5). Thus the complete deformation corresponding to the $r$-matrix $r_{3}$ is the twisted deformation of $U_{r^{\prime}_{3}}(\mathfrak{o}(3,1))$, i.e. the resulting coproduct $\Delta_{r_{3}}$ is given as follows

[TABLE]

and in this case the resulting antipode $S_{r_{3}}$ does not change, $S_{r_{3}}=S_{r^{\prime}_{3}}$. Applying the twisting two-tensor (3.33) to the formulas (3.24) and (3.25) we obtain

[TABLE]

Next, we describe quantum deformation corresponding to the classical $r$-matrix $r_{4}$ (3.21). Since the $r$-matrix $r_{4}^{\prime}(\alpha):=r_{4}^{\prime}$ is a particular case of $r_{3}(\alpha,\beta,\gamma):=r_{3}$, namely $r_{4}^{\prime}(\alpha)=r_{3}(\alpha,\beta=0,\gamma=\alpha)$, therefore a quantum deformation corresponding to the $r$-matrix $r_{4}^{\prime}$ is obtained from the previous case by setting $\beta=0,\gamma=\alpha$, and we have the following formulas for the coproducts $\Delta_{r_{4}^{\prime}}$:

[TABLE]

where we set $\xi:=\imath\alpha$.

Consider the two-tensor

[TABLE]

Using properties of $q$-exponentials (see [17]) is not hard to verify that $F_{r_{4}^{\prime\prime}}$ satisfies the cocycle equation (2.4). Thus the quantization corresponding to the $r$-matrix $r_{4}$ is the twisted $q$-deformation $U_{r_{4}^{\prime}}(\mathfrak{o}(3,1))$. Explicit formulas of the co-products $\Delta_{r_{4}}(\cdot)=F_{r_{4}^{\prime\prime}}\Delta_{r_{4}^{\prime}}(\cdot)F_{r_{4}^{\prime\prime}}^{-1}$ and antipodes $S_{r_{4}}(\cdot)$ in the complex and real Cartan-Weyl bases of $U_{r_{4}^{\prime}}(\mathfrak{o}(3,1))$ will be presented in the outgoing paper [7].

4 Quantum deformations of Poincare algebra

The Poincaré algebra ${\mathcal{P}}(3,1)$ of the 4-dimensional space-time is generated by 10 elements: the six-dimensinal Lorentz algebra $\mathfrak{o}(3,1)$ with the generators $M_{i}$, $N_{i}$ ($i=1,2,3$):

[TABLE]

and the four-momenta $P_{j}$, $P_{0}$ $(j=1,2,3)$ with the standard commutation relations:

[TABLE]

The physical generators of the Lorentz algebra, $M_{i}$, $N_{i}$ ($i=1,2,3$), are related with the canonical basis $h,h^{\prime},e_{\pm},e^{\prime}_{\pm}$ as follows

[TABLE]

The subalgebra generated by the four-momenta $P_{j}$, $P_{0}$ $(j=1,2,3)$ will be denoted by $\mathbf{P}$ and we also set $P_{\pm}:=P_{0}\pm P_{3}$.

S. Zakrzewski has shown in [2] that each classical $r$-matrix, $r\in\mathcal{P}(3,1)\wedge\mathcal{P}(3,1)$, has a decomposition

[TABLE]

where $a\in\mathbf{P}\wedge\mathbf{P}$, $b\in\mathfrak{o}(3,1)\wedge\mathbf{P}$, $c\in\mathfrak{o}(3,1)\wedge\mathfrak{o}(3,1)$ satisfy the following relations

[TABLE]

Here $[[\cdot,\cdot]]$ means the Schouten bracket. Moreover a total list of the classical $r$-matrices for the case $c\neq 0$ and also for the case $c=0$, $t=0$ was found.999Classification of the $r$-matrices for the case $c=0$, $t\neq 0$ is an open problem up to now. It was shown that there are fifteen solutions for the case $c=0$, $t=0$, and six solutions for the case $c\neq 0$ where there is only one solution for $t\neq 0$. Thus Zakrzewski found twenty $r$-matrices which satisfy the homogeneous classical Yang-Baxter equation ($t=0$ in (4.9)). Analysis of these twenty solutions shows that each of them can be presented as a sum of subordinated $r$-matrices which almost all are of Abelian and Jordanian types. Therefore these $r$-matrices correspond to twisted deformations of the Poincaré algebra ${\mathcal{P}}(3,1)$. We present here $r$-matrices only for the case $c\neq 0,\;t=0$:

[TABLE]

The first $r$-matrix $r_{1}$ is a sum of two subordinated Abelian $r$-matrices

[TABLE]

Therefore the total twist defining quantization in the direction to this $r$-matrix is the ordered product of two the Abelian twits

[TABLE]

The second $r$-matrix $r_{2}$ is a sum of three subordinated $r$-matrices where two of them are of Abelian type and one is of Jordanian type

[TABLE]

Corresponding twist is given by the following formulas

[TABLE]

where

[TABLE]

Here and below we set $\sigma_{+}:=\frac{1}{2}\ln(1+\beta_{1}P_{+})$.

The third $r$-matrix $r_{3}$ is a sum of two subordinated $r$-matrices where one is of Abelian type and another is a more complicated $r$-matrix which we call mixed Jordanian-Abelian type

[TABLE]

Corresponding twist is given by the following formulas

[TABLE]

where

[TABLE]

The fourth $r$-matrix $r_{4}$ is a sum of two subordinated $r$-matrices of Abelian type

[TABLE]

Corresponding twist is given by the following formulas

[TABLE]

where

[TABLE]

The fifth $r$-matrix $r_{5}$ is the $r$-matrix of the Lorentz algebra, (3.6), and the corresponding twist is given by the formula (3.13).