Lie Superalgebras Generated by Reflections in Weyl Groups of Classical Type

Christopher M. Drupieski; Jonathan R. Kujawa

doi:10.5802/art.45

paper

Lie Superalgebras Generated by Reflections in Weyl Groups of Classical Type

arXiv:2509.00945 · doi:10.5802/art.45

Abstract

We consider the finite Weyl groups of classical type -- $W (A_{r})$ for $r \geq 1$ , $W (B_{r}) = W (C_{r})$ for $r \geq 2$ , and $W (D_{r})$ for $r \geq 4$ -- as supergroups in which the reflections are of odd superdegree. Viewing the corresponding complex group algebras as Lie superalgebras via the graded commutator bracket, we determine the structure of the Lie sub-superalgebras generated by the sets of reflections. In each case, this Lie superalgebra is equal to the full derived subalgebra of the group algebra plus the span of the class sums of the reflections.

arXiv comment33 pages. Version 2 includes various minor corrections and improvements to the exposition. The GAP code used in the paper can be obtained by downloading the full set of source files from the ArXiv