Translated simple modules for Lie algebras and simple supermodules for Lie superalgebras

Abstract

We prove that the tensor product of a simple and a finite dimensional sln-module has finite type socle. This is applied to reduce classification of simple q(n)-supermodules to that of simple sln-modules. Rough structure of simple q(n)-supermodules, considered as sln-modules, is described in terms of the combinatorics of category O.MSC 2010: 17B10 17B55

Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras

“…It follows that the module Ind P 0 (λ) ∼ = Ind g g ≥0 Ind g ≥0 g 0 P 0 (λ), admits a Kac flag, subquotients of which are of the form K(P 0 (γ)), where we have γ ∈ Λ(ν) >λ ∪ {λ}. By [CCM,Theorem 51], each K(P 0 (γ)) has simple top, which is, automatically, isomorphic to L(γ). Consequently, K(P 0 (γ)) is a quotient of P (γ).…”

On semisimplicity of Jantzen middles for the periplectic Lie superalgebra

“…To see this, we consider the following short exact sequence 0 → L(λ) → M (ω 2 ) → K(L(ω 2 )) → 0, (3.4) obtained by applying the Kac functor K(−) to the short exact sequence 0 → L(λ) → M (ω 2 ) → L(ω 2 ) → 0. Since the socle of K(L(ω 2 )) is isomorphic to L(0) and M (ω 2 ) has simple socle (see, e.g., [CCM,Theorem 51]) isomorphic to L(λ), we may conclude that Ext 1 O ( L(0), L(λ)) = 0. By Proposition 5, it follows that L(0) is isomorphic to a submodule of U α (λ).…”

“…Next, we show that Ext 1 O ( L(µ), L(λ)) = 0, which will imply that [socU α (λ) : L(µ)] = 0 by Proposition 5. We recall that every Verma module has a simple socle by [CCM,Theorem 51]. Applying the Kac functor K(−) to the short exact sequence 0 → L(λ) → M (s • λ) → L(s • λ) → 0, we then obtain a non-split short exact sequence 0 → L(λ) → M (s • λ) → K(L(s • λ)) → 0, and so L(λ) = soc M (s • λ).…”

Whittaker modules for classical Lie superalgebras

“…Let λ ∈ W • λ be antidominant. By [CCM,Theorem 51] and Lemma 25 the socle of M (λ) is isomorphic to the socle of K(λ), which is a simple module of antidominant highest weight γ with λ = γ; see also [CCC,Theorem 4.4,Proposition 4.15]. Using the grading operator d g from [CM,Sections 5.1,5. The Cartan subalgebra h ⊂ g 0 consists of diagonal matrices above.…”