Is there a definition of analogue Weyl group for Lie super algebra?

Question

I heard from some people working in Lie super algebra that there was no proper definition for Weyl group of Lie super algebra. I do not know Lie super algebra at all. But When I searched on Google, I found that it seems there still exists some definitions of Weyl group.

I wonder whether there is a well-accepted definition for it.

The reason I want to ask this question is that I need Weyl group for Lie super algebra to play with the geometry related to super Lie algebra.

Another question is that I heard from some experts in Lie super algebra that there was no well-accepted super geometry related to Lie super algebra.

However, It seems that one of the students of Manin, who is Dimitry Leites ever developed supergeometry.

Answer 1

From the point of view of super geometry, Manin and I introduced the notion of a super Weyl group in relation to the geometry of homogeneous superspaces. We constructed Schubert supercells which were labeled by elements of a super Weyl group. I am not sure whether this is a well-accepted definition; it was rather a construction done in an ad hoc way for each classical simple Lie supergroup. The results were announced in

Manin, Y. I.; Voronov, A. A. Schubert supercells. Functional Anal. Appl. 18, 329-330 (1985).

Voronov, A. A. Relative disposition of the Schubert supervarieties and resolution of their singularities. Functional Anal. Appl. 21, 62-64 (1987).

and published in detail in

Manin, Y. I.; Voronov, A. A. Supercellular partitions of flag superspaces. Current problems in mathematics. Newest results, USSR Acad. Sci., Moscow. 32, 27-70 (1988). (in Russian). English translation: J. Soviet Math. 51(1), 2083-2108 (1990).

Answer 2

I'm not an expert in this area, but I'm told that the key phrase in the superalgebra world is "Weyl groupoid" rather than Weyl group. I did not look at the construction long enough to understand it. Serganova has a paper describing foundations in a super analogue of the Kac-Moody setting, and you can find a description of the Weyl groupoid there.

Edit: The situation seems to be the following: For Kac-Moody algebras, there is a unique conjugacy class of Cartan subalgebra (under automorphisms), and the Weyl group acts transitively on systems of simple roots. These properties fail to hold in the superalgebra setting. One instead can form a groupoid whose objects are finite size square matrices $A$ with integer entries (or the Lie superalgebras $g(A)$ obtained by a generators-and-relations construction), and whose morphisms from $A$ to $A'$ are superalgebra isomorphisms $g(A) \to g(A')$ that take a Cartan of $g(A)$ to a Cartan of $g(A')$. The Weyl groupoid of $g(A)$ is then the connected component of $A$ in the larger groupoid.

Regarding geometry, I think Penkov has done some work with flag supermanifolds and Borel-Weil-Bott. I don't think there is much debate about the foundations of supermanifold theory, but I guess the geometric representation theory doesn't extend from the even case by rote translation of proofs.

Answer 3

The answer to the question in the title is affirmative. In the Dictionary of Lie superalgebras, there is an entry on the Weyl group of a classical Lie superalgebra. It is generated by reflections associated to the simple even roots, hence it is the standard Weyl group of the even subalgebra. In addition, they also mention that one can extend the Weyl group by the addition of so-called generalised Weyl transformations associated to the odd roots. They also give a couple of references.

As for the geometry associated to Lie superalgebras, there is a notion of Lie supergroup (this link is the not-particularly-good wikipedia article), which stands in relation to Lie superalgebras just as their non-super counterparts. Lie supergroups are particular examples of supermanifolds, on which there is a substantial literature.

Answer 4

I believe that it is important to mention the notion of Weyl groupoid.

The abstract Weyl groupoid was defined by Heckenberger and Yamane. (In their paper you will find some basic material about Weyl groupoids, generalized root systems and some ideas about the role played by the Weyl groupoid in Lie superalgebras.) Two years later, in 2008, Cuntz and Heckenberger reformulated the definition of Weyl groupoids in terms of Cartan schemes.

Here you will find an interesting mini-course on Weyl groupoids.

Some references:

• Cuntz, M.; Heckenberger, I. Weyl groupoids with at most three objects. J. Pure Appl. Algebra 213 (2009), no. 6, 1112--1128. MR2498801 (2010b:20100), link, arXiv
• Heckenberger, István; Yamane, Hiroyuki. A generalization of Coxeter groups, root systems, and Matsumoto's theorem. Math. Z. 259 (2008), no. 2, 255--276. MR2390080 (2009e:20087), link, arXiv

Answer 5

As it is clear from the other answers there are several viewpoints on Weyl group in super case.

Let me mention papers by Sergeev and Veselov, who also stands on the point of the Weyl groupoid in the super case. As far as I understand such viewpoint agrees with their works on Calogero-Moser integrable system.

https://arxiv.org/abs/1504.08310 "Orbits and invariants of super Weyl groupoid"

https://arxiv.org/abs/0704.2250 "Grothendieck rings of basic classical Lie superalgebras"

Abstract:

"The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalised root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called Weyl groupoids."

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PS

By the way, GL(n,F_q) for q=1 "is" Weyl group (see e.g. field with one element). I wonder is there some kind of super analog of that heuristics ? Is there some kind of "bijection" between "irreps" and "conjugacy classes" for Weyl grouppoid (see MO270916) ?