On modified extension graphs of a fixed atypicality

Abstract

In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras gl(m|n), osp(M |2n) and q m . We consider a simplified version of the extension graph which is produced from the Ext 1 -graph by identifying representations obtained by parity change and removal of the loops. We give a necessary condition for a pair of vertices to be connected and show that this condition is sufficient in most of the cases. This condition implies that the image of a finite-dimensional simple module under the Duflo-Serganova functor has indecomposable isotypical components. This yields semisimplicity of Duflo-Serganova functor for Fin(gl(m|n)) and for Fin(osp(M |2n)).

On the Duflo-Serganova functor for the queer Lie superalgebra

“…The formula (d ′ ) follows from Theorem 5.1 (i) and the following fact obtained in [13]: if Ext 1 (L(ν), L(ν ′ )) = 0 for some distinct dominant weights ν, ν ′ , then the diagram of one for these weights is obtained from the diagram of the other one by moving a symbol × along one of the arches (where ∧ is considered as a half of ×). The same reasoning works in the Kac-Moody case.…”