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Representations of shifted super Yangians and finite $W$-superalgebras of type A

Kang Lu, Yung-Ning Peng

Abstract

In this article, we study the representation theory of shifted super Yangians and finite $W$-superalgebras of type A. A criterion for the finite dimensionality of irreducible modules is obtained in the standard parity case. Furthermore, we provide an explicit Gelfand-Tsetlin character formula for Verma modules of finite $W$-superalgebras. As an application, we show that the centers of the finite $W$-superalgebras associated to any even nilpotent elements belonging to the same general linear Lie superalgebra are all isomorphic to the center of the universal enveloping superalgebra.

Representations of shifted super Yangians and finite $W$-superalgebras of type A

Abstract

In this article, we study the representation theory of shifted super Yangians and finite -superalgebras of type A. A criterion for the finite dimensionality of irreducible modules is obtained in the standard parity case. Furthermore, we provide an explicit Gelfand-Tsetlin character formula for Verma modules of finite -superalgebras. As an application, we show that the centers of the finite -superalgebras associated to any even nilpotent elements belonging to the same general linear Lie superalgebra are all isomorphic to the center of the universal enveloping superalgebra.
Paper Structure (31 sections, 50 theorems, 203 equations)

This paper contains 31 sections, 50 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Shifted super Yangians
  3. Shifted super Yangians
  4. PBW basis
  5. Properties of shifted Yangians
  6. Some isomorphisms
  7. $\mathbf Q_{m|n}$-grading
  8. Coproduct
  9. Quantum Berezinian and center of ${\mathrm{Y}_{m|n}(\sigma)}$
  10. Finite $W$-superalgebras
  11. Pyramid
  12. Finite $W$-superalgebras
  13. Identification
  14. Parabolic induction
  15. Harish-Chandra homomorphisms
  16. ...and 16 more sections

Key Result

Theorem A

The restriction of the coproduct $\Delta:\mathrm{Y}_{m|n}\rightarrow \mathrm{Y}_{m|n} \otimes \mathrm{Y}_{m|n}$ induces a homomorphism Moreover, this homomorphism factors through the corresponding quotients to yield a homomorphism of finite $W$-superalgebras

Theorems & Definitions (95)

  • Theorem A: Theorem \ref{['thm:grownup']}, Corollary \ref{['SWcop']}
  • Theorem B: Theorem \ref{['thm:fd']}
  • Theorem C: Theorem \ref{['thm:fd-W']}, Theorem \ref{['thm:character']}
  • Theorem D: Theorem \ref{['isocent']}
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4: Pe21
  • Lemma 2.5
  • proof
  • ...and 85 more