Lie Superalgebras Generated by Reflections in Weyl Groups of Classical Type

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.