Highest weight modules for affine Lie superalgebras

Abstract

We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.

NON-STANDARD VERMA TYPE MODULES FOR 𝔮(n)(2)

“…Compare also with the isotropic case of [CF18]. (b) Heisenberg algebras admit a family of triangular decompositions parametrized by maps ϕ : N → {±} d , where d is a certain dimension.…”

“…Applying Theorem 4.7 in the case X = ∅ gives: Remark 4.9. Notice that differently from the other cases studied in the literature, we do not need the central charge to be nonzero in order to have M(ĝ, m; L( m, λ)) to be irreducible (compare with [Cox94,Fut94,CF18]). This is due to the fact that the central element K does not play a role in the action of the imaginary subalgebra H on L( m, λ).…”

“…Affine symmetrizable superalgebras were described in [Ser11] and [vdL89]. Theory of Verma type modules for affine Lie superalgebras was developed in [ERF09] and [CF18]. In particular, given a Borel subsuperalgebra b of the affine Lie superalgebra ĝ and a 1-dimensional representation C λ of b for some weight λ of the Cartan subalgebra of ĝ, one can construct the Verma type module…”

“…The Verma type module is non-standard if b does not contain all positive root subspaces for some basis of the root system of ĝ. In the case the finite-dimensional Lie superalgebra associated to ĝ is a contragredient Lie superalgebra, all Borel subsuperalgebras of ĝ were described in [CF18], see also [DFG09]. The paper [CF18] also gives a criterion for the irreducibility for non-standard Verma type module.…”