Root groupoid and related Lie superalgebras
Abstract
We introduce a notion of a root groupoid as a replacement of the notion of Weyl group for (Kac-Moody) Lie superalgebras. The objects of the root groupoid classify certain root data, the arrows are defined by generators and relations. As an abstract groupoid the root groupoid has many connected components and we show that to some of them one can associate an interesting family of Lie superalgebras which we call root superalgebras. We classify root superalgebras satisfying some additional assumptions. To each root groupoid component we associate a graph (called skeleton) generalizing the Cayley graph of the Weyl group. We establish the Coxeter property of the skeleton generalizing in this way the fact that the Weyl group of a Kac-Moody Lie algebra is Coxeter.
Cite
@article{arxiv.2209.06253,
title = {Root groupoid and related Lie superalgebras},
author = {Maria Gorelik and Vladimir Hinich and Vera Serganova},
journal= {arXiv preprint arXiv:2209.06253},
year = {2024}
}
Comments
4th version: accepted to Annals in Representation Theory. Warning: the numbering of subsections and theorems in the arXiv version has remained as we like it, whereas it has changed according to the journal's requirements in the Journal version