On primitive axial algebras of Jordan type
Jonathan I. Hall, Yoav Segev, and Sergey Shpectorov

TL;DR
This paper reviews primitive axial algebras of Jordan type, explores their connections with 3-transposition groups and Matsuo algebras, and proves the existence of Frobenius forms for these algebras across all parameters.
Contribution
It provides an overview of the current understanding and establishes that primitive axial algebras of Jordan type admit Frobenius forms for any parameter.
Findings
Primitive axial algebras of Jordan type half are connected to 3-transposition groups.
Matsuo algebras are related to these axial algebras.
Frobenius forms exist for primitive axial algebras of Jordan type for all parameters.
Abstract
In this note we give an overview of our knowledge regarding primitive axial algebras of Jordan type half and connections between -transposition groups and Matsuo algebras. We also show that primitive axial algebras of Jordan type admit a Frobenius form, for any .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
On primitive axial algebras of Jordan type
J.I. Hall Y. Segev S. Shpectorov
Jonathan, I. Hall
Department of Mathematics
Michigan State University
Wells Hall, 619 Red Cedar Road, East Lansing, MI 48840
United States
Yoav Segev
Department of Mathematics
Ben-Gurion University
Beer-Sheva 84105
Israel
Sergey Shpectorov
School of Mathematics
University of Birmingham
Watson Building, Edgbaston, Birmingham, B15 2TT
United Kingdom
Dedicated to Professor Robert L. Griess, Jr. on the occasion of his st birthday
(Date: May 9, 2017)
Abstract.
In this note we give an overview of our knowledge regarding primitive axial algebras of Jordan type half and connections between -transposition groups and Matsuo algebras. We also show that primitive axial algebras of Jordan type admit a Frobenius form, for any .
Key words and phrases:
Axial algebra, 3-transposition, Jordan algebra, Frobenius form
2010 Mathematics Subject Classification:
Primary: 17A99; Secondary: 17C99, 17B69
1. Introduction
The purpose of this note is threefold. In §2 we give an overview of our knowledge regarding primitive axial algebras of Jordan type half. This is taken from [HSS]. In fact we focus in §2 on one of the main results in [HSS] which characterizes Jordan algebras of Clifford type amongst primitive axial algebras of Jordan type half. The primitive axial algebras of Jordan type are reviewed (amongst other things) by Jon Hall in another paper of this volume. In §3, we complete, for the case a result connecting -transposition groups and Matsuo algebras, established in [HRS, Theorem 6.3] for . In §4 we show that any primitive axial algebra of Jordan type (any ) admits a Frobenius form.
We start by recalling a few definitions. We do not give the historical background as it can be best found in the introduction to [HRS].
All algebras in this note are commutative, non-associative over a field of characteristic not .
For the adjoint operator is multiplication by , so
[TABLE]
An axis in is, by definition, a semisimple idempotent, i.e., an idempotent whose minimal ad-polynomial has few distinct linear factors; where the minimal ad-polynomial is the minimal polynomial of the linear operator (we are not assuming that is finite dimensional, however, we are assuming that has a minimal polynomial).
Axial algebras, introduced recently by Hall, Rehren and Shpectorov ([HRS]), are, by definition, algebras generated by axes. When certain fusion rules, i.e. multiplication rules, between the eigenspaces corresponding to an axis, are imposed the structure of axial algebras remains interesting yet it is more rigid.
Given an element and a scalar the -eigenspace of is denoted so:
[TABLE]
(We allow .)
Axial algebras of Jordan type where is fixed, are algebras generated by a set of axes such that for each
- (1)
The minimal ad-polynomial of divides . 2. (2)
The fusion rules imitate the Peirce multiplication rules in Jordan algebras. These fusion rules are:
[TABLE]
[TABLE]
[TABLE]
In particular, if we set
[TABLE]
then
[TABLE]
for
Thus, for example, Jordan algebras are axial algebras of Jordan type provided that they are generated by idempotents.
An axis is absolutely primitive if (this is stronger than the usual notion of primitivity). We call an absolutely primitive axis satisfying (1), (2) above an -axis.
A primitive axial algebra of Jordan type is an algebra generated by -axes. For primitive axial algebras of Jordan type were thoroughly analyzed by Hall, Rehren, and Shpectorov in [HRS]. The case , is much less understood and is of a different nature. This case is the focus of [HSS] and of §§2,3 of this note.
Given an -axis recall that
[TABLE]
The map defined by where is an automorphism of of order or . It is called the Miyamoto involution corresponding to .
1.1. Jordan algebras of Clifford type
A Jordan algebra of Clifford type consists of the following information:
- (1)
A vector space over together with a symmetric bilinear form on . The corresponding quadratic form is denoted . 2. (2)
The Jordan algebra is with multiplication defined by
[TABLE]
The algebra comes from the associative Clifford algebra : it is a sub-Jordan algebra of where, as usual, denotes the special Jordan algebra that emerges from the associative algebra
Let . It is easy to check that:
- (a)
For and , the element is an idempotent if and only if and .
- (b)
Assume that is an idempotent in . Then
- (i)
so is a -axis. (Thus is a primitive axial algebra of Jordan type iff it is generated by idempotents.)
- (ii)
(of course is a -axis), and
- (iii)
where .
- (c)
It follows that for any -axis .
The purpose of §2 is to show that property (c) above essentially characterizes Jordan algebras of Clifford type amongst primitive axial algebras of Jordan type .
2. Primitive axial algebras of Jordan type half
Throughout this section is a primitive axial algebra of Jordan type generated by a set of -axes.
Let be the graph on the set of all -axes of where distinct form an edge iff . Let also be the full subgraph of on the set . The purpose of this section is to sketch a proof of the following theorem:
Theorem 2.1**.**
Assume that is connected and that there are two distinct -axes such that . Then is the identity of and is a Jordan algebra of Clifford type.
In the remainder of this section we will sketch a proof of Theorem 2.1. First we need a theorem that enables us to identify as a Jordan algebra of Clifford type in the case .
Theorem 2.2**.**
Let . Assume that contains two -axes such that and such that for all where . Then is a Jordan algebra of Clifford type.
We do not include a proof of Theorem 2.2, see [HSS, Theorem 5.4].
We will need some information about -generated subalgebras of . This information is taken from [HRS]. Let with . We denote by ** the subalgebra generated by and If contains an identity element, we denote it by .** Note that by [HRS], -generated subalgebras are at most -dimensional.
Lemma 2.3** (Lemma 3.1.2 in [HSS]).**
Let with . Then is -dimensional precisely in the following cases:
- (1)
* we then denote: .* 2. (2)
* we then denote: .* 3. (3)
* we then denote: .*
Furthermore,
- (4)
the algebras in cases and above do not have an identity element.
The following proposition deals with -generated -dimensional subalgebras.
Proposition 2.4** (Proposition 4.6 [HRS]).**
Let with . Then is -dimensional precisely when and there exists and a scalar such that if we set then
- (1)
** 2. (2)
* for all *
furthermore
- (3)
* contains an identity element if and only if in which case .*
When is -dimensional we denote: where is the scalar mentioned above.
From now on we assume that ** is connected**. Note that by [HSS, Lemma 6.4], is connected iff is connected. Further, we assume that ** are distinct with **.
Proposition 2.5** (Proposition 6.5 in [HSS]).**
* and*
- (1)
for any exactly one the following holds:
- (i)
**
- (ii)
* and for some we have is -dimensional, and . Further .*
- (iii)
* is -dimensional and .* 2. (2)
If is an -axis in such that then .
Proof sketch.
By [HSS, Lemma 3.2.1], for any we have and since, by definition, we see that .
If then, as above (and vice versa), so (i) holds. Hence we may assume that .
If then by [HRS, Proposition 6.5], and since is connected, a contradiction. Thus .
Now consider
[TABLE]
is either or -dimensional. If is -dimensional, then and since one shows that ([HSS, Lemma 3.2.5]), so (iii) holds.
So suppose is -dimensional. If both and are -dimensional, then they both equal to . But then or a contradiction.
Therefore without loss is -dimensional and is -dimensional. If then (ii) holds: Clearly and and then a careful analysis of the situation gives (ii).
The case where both and are -dimensional and is -dimensional is the hardest case and some precise work is required to get a contradiction. ∎
Proposition 2.6**.**
* and*
- (1)
* for all * 2. (2)
* contains an identity element * 3. (3)
for any such that is -dimensional we have .
Proof.
Let be the distance function on . Let
[TABLE]
Since is connected . Also, by Proposition 2.5(1i), . Let . By Proposition 2.5, and after perhaps interchanging and is -dimensional and . Set
[TABLE]
then
[TABLE]
Let be at distance from in and let
[TABLE]
Without loss is -dimensional and . Now
- •
- •
because
- •
so
- •
so
- •
W:={\rm Span}\big{(}\{y,\ y^{\tau(x)}\}\big{)}\cap{\rm Span}\big{(}\{x,\ x^{\tau(y)}\}\big{)}\neq\{0\}. Indeed, is the intersection of two -dimensional subspaces of which is of dimension at most .
- •
both annihilates and acts as identity on a contradiction.
Hence and clearly in . But now, as we saw above, for all . It follows that is the identity of and (3) holds as well. ∎
We are now in a position to prove Theorem 2.1.
Proof of Theorem 2.1.
We show that the hypotheses of Theorem 2.2 are satisfied. By Proposition 2.6, and . Let . Then
[TABLE]
Clearly if . Otherwise, by Proposition 2.6(1), . If is -dimensional, then since and so . If is -dimensional, then by Proposition 2.6(3), . Furthermore by [HRS], for some and again . ∎
3. -transpositions and Matsuo algebras
Recall that a set of axes is closed iff for all . In this section is a primitive axial algebra of Jordan type generated by a closed set of -axes such that .
Let be a group generated by a normal set of involutions . Recall that is called a set of -transpositions in if for all . The group is then called a -transposition group.
Let be a normal set of -transpositions in the group that generate . The Matsuo algebra associated with the pair denoted here is defined as follows. As a vector space over it has the basis . Multiplication is defined for as follows
[TABLE]
This is extended by linearity to the entire algebra. (Note that we denote multiplication in by juxtaposition and in by dot.) By [HRS, Theorem 6.2], is a primitive axial algebra of Jordan type .
The purpose of this section is to prove the following Theorem:
Theorem 3.1**.**
Suppose that the graph is connected. Let and . Assume that the map on is injective and that is a set of -transpositions in . Then is a quotient of the Matsuo algebra .
Remark 3.2**.**
Theorem 3.1 was proved in [HRS, Theorem 6.3] for . The proof for needed a correction, in view of [HSS]. Note that the summand does not appear in Theorem 3.1 since we are assuming that is connected. We also mention that for the map on defined by is always injective, by [HSS, Proposition 6.5], and since is connected.
We included a proof of Theorem 3.1 for all for completeness.
Lemma 3.3**.**
* for all .*
Proof.
Clearly this holds when so assume . Suppose first that is -dimensional. We use [HSS, Lemma 3.1.2]. If then and (see also [HSS, Lemma 3.2.1]), so the claim holds.
Suppose next that . Then and (see also [HSS, Lemma 3.1.8]), so the claim holds.
Assume that . Then and (see also [HSS, Lemma 3.1.9]), so again the claim holds.
We may assume that is -dimensional. Set . By[HSS, Theorem 3.1.3(6)], . Also, . Hence we get
[TABLE]
Corollary 3.4** (See Corollary 1.2 in [HRS]).**
* is spanned over by .*
Proof.
This is immediate from Lemma 3.3 and the definition of a closed set of axes. ∎
Lemma 3.5**.**
Suppose that
[TABLE]
Let be distinct. Then
- (1)
if then . 2. (2)
if then .
Proof.
(1): By [HSS, Lemmas 3.2.7(2) and 3.1.6(2)] and by so (1) holds (see also [HSS, Lemma 3.1.2(1a)]).
(2): If then (2) follows from [HRS, Proposition 4.8]. So suppose . By [HSS, Lemma 3.2.7(1) and Corollary 3.3.2] and by we get ∎
We can now prove Theorem 3.1.
Proof of Theorem 3.1.
Set . We claim that the map
[TABLE]
extended by linearity is a surjective algebra homomorphism. Note that is well defined since the map is injective on .
Now is surjective by Corollary 3.4. Next we need to check that
[TABLE]
If then , and so holds.
If then while by Lemma 3.5(1), so holds in this case as well.
Finally assume that . Then
[TABLE]
where the last equality follows from the standard fact that . Thus . However, by Lemma 3.5(2) and Lemma 3.3, so holds in this case as well and the proof of the theorem is complete. ∎
4. The existence of a Frobenius form
Recall that a non-zero bilinear form on an algebra is called Frobenius if the form associates with the algebra product, that is,
[TABLE]
for all .
For primitive axial algebras of Jordan type we specialize the concept of Frobenius form further by asking that the condition be satisfied for each -axis .
The purpose of this section is to prove the following theorem:
Theorem 4.1**.**
Let be a primitive axial algebra of Jordan type . Then admits a Frobenius form.
The proof of Theorem 4.1 depends on two properties of primitive axial algebras of Jordan type. The first is Corollary 3.4. The second is proven in [HRS] (Lemma 4.2 below).
For an -axis , let be the projection function with respect to . That is, for , we have that , where and are eigenvectors of the adjoint linear transformation for the eigenvalues [math] and , respectively.
Lemma 4.2** (Lemma 4.4 in [HRS]).**
For a primitive axial algebra of Jordan type and for any -axes , we have .
Note that the constant that we used earlier for -axes is the same as .
Proof of Theorem 4.1.
We start by defining the bilinear form on . Using Corollary 3.4 we can select a basis of consisting of -axes, and we let
[TABLE]
Extending by linearity we get the bilinear form . Note that Lemma 4.2 implies that is symmetric.
Lemma 4.3**.**
- (1)
* for all -axes and all * 2. (2)
* for all -axes * 3. (3)
* is invariant under automorphisms of .*
Proof.
(1&2): Let be an -axis and suppose that
[TABLE]
Since is linear,
[TABLE]
and (1) holds for . Now if then holds by definition, so (1) holds for . Suppose . Let . Then by Lemma 4.2, and as (1) holds for . Finally, since is symmetric so and holds for any -axis . This shows that (1) holds.
In particular, for every -axis , we have that , since, clearly, . Thus (2) holds.
(3): Let , if is the decomposition of with respect to the -axis then is the decomposition of with respect to the -axis . Hence , and so . Finally, taking an arbitrary and decomposing it with respect to the basis as , we get that . So indeed, is invariant under the automorphisms of . ∎
Lemma 4.4**.**
For every -axis , different eigenspaces of are orthogonal with respect to .
Proof.
Clearly, if then . Hence is orthogonal to both and . It remains to show that these two are also orthogonal to each other. Let and , the fact that is invariant under gives us . Clearly, this means that . ∎
We are now ready to complete the proof that associates with the algebra product. Note that the identity
[TABLE]
that we need to prove is linear in , , and . In particular, since is spanned by -axes, we may assume that is an -axis. Furthermore, since decomposes as the sum of the eigenspaces of , we may assume that and are eigenvectors of , say, for the eigenvalues and . We have two cases:
If then
[TABLE]
If then
[TABLE]
since and are orthogonal to each other. Thus, in both cases we have the desired equality , proving that the form is Frobenius. ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[HRS] J.I. Hall, F. Rehren, S. Shpectorov, Primitive axial algebras of Jordan type, J. Algebra 437 (2015), 79–115.
- 2[HSS] J.I. Hall, Y. Segev, S. Shpectorov, Miyamoto involutions in axial algebras of Jordan type half, to appear in Israel J. Math.
