Whittaker Categories and Whittaker Modules for Lie Superalgebras

Abstract

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, a description of the strongly typical simple Whittaker modules.

Whittaker modules for classical Lie superalgebras

“…The sets Φ + 0 , Φ + 1 are given as follows: In this subsection, we will construct composition series of standard Whittaker modules of N (ζ) explicitly by finding their Whittaker vectors. Similar computation was also given in [BCW,Section 5.1], where the authors concluded that all standard Whittaker modules are simple. However, by [CM,Theorem 6.7] and [MS1,Proposition 2.1(3)] there exist reducible standard Whittaker modules; see Section 5.3.3 for more details.…”

“…We should mention that the Part (1) in Theorem 8 generalizes the construction of simple modules in [BCW,Section 4], where the case of some basic classical Lie superalgebras of type I was considered.…”

“…Following [Ko], elements in Wh ζ (V ) and Wh ζ (M ) are called Whittaker vectors. The following useful lemma is taken from [BCW,Lemma 3.3].…”

“…However, by [CM,Theorem 6.7] and [MS1,Proposition 2.1(3)] there exist reducible standard Whittaker modules; see Section 5.3.3 for more details. It is worth pointing out that the assumption of typical weight λ is needed to be added in the calculation in [BCW,Section 5.1].…”

Whittaker modules for the super-Virasoro algebras

“…However, there are several differences and some features that are new in the super case. It has been observed in [5], Mathematics Subject Classification: 17B65; 17B68;17B70. 1 simple finite-dimensional modules for a finite-dimensional nilpotent Lie superalgebra are not always one-dimensional [25].…”