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Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras

Chih-Whi Chen, Shun-Jen Cheng, Uhi Rinn Suh

Abstract

For a basic classical Lie superalgebra $\mathfrak s$, let $\mathfrak g$ be the central extension of the Takiff superalgebra $\mathfrak s\otimesΛ(θ)$, where $θ$ is an odd indeterminate. We study the category of $\mathfrak g$-Whittaker modules associated with a nilcharacter $χ$ of $\mathfrak g$ and show that it is equivalent to the category of $\mathfrak s$-Whittaker modules associated with a nilcharacter of $\mathfrak s$ determined by $χ$. In the case when $χ$ is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite $W$-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite $W$-superalgebra associated to $\mathfrak s$. Here, a supersymmetric finite $W$-algebra is conjecturally the Zhu algebra of a supersymmetric affine $W$-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric $W$-algebra.

Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras

Abstract

For a basic classical Lie superalgebra , let be the central extension of the Takiff superalgebra , where is an odd indeterminate. We study the category of -Whittaker modules associated with a nilcharacter of and show that it is equivalent to the category of -Whittaker modules associated with a nilcharacter of determined by . In the case when is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite -algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite -superalgebra associated to . Here, a supersymmetric finite -algebra is conjecturally the Zhu algebra of a supersymmetric affine -algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric -algebra.

Paper Structure

This paper contains 23 sections, 25 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Description of results
  4. Preliminary
  5. Basic classical Lie superalgebras
  6. Central extensions
  7. Representation Theory
  8. Category $\mathcal{O}$
  9. Irreducible $\hat{\mathfrak{h}}$-modules
  10. Verma and simple modules
  11. Block decomposition
  12. Tensor product of simple modules
  13. Fock space $\mathfrak F_c$
  14. Whittaker modules
  15. Whittaker categories
  16. ...and 8 more sections

Key Result

Theorem 1

For an arbitrary character $\chi: \hat{\mathfrak n}_{\bar{0}} \rightarrow \mathbb C$, the tensor functor is an equivalence of categories. In particular, if $\chi$ vanishes on $\mathfrak n_{\bar{0}}$, then the categories $\mathfrak{g}\emph{-Wmod}^{\chi}_c$ and $\mathfrak{g}\emph{-Wmod}^{\chi}_{c'}$ are equivalent, for all $c,c'\not=0$.

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • ...and 36 more