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Twisted super Yangians of type AIII and their representations

Kang Lu

Abstract

We study the super analogue of the Molev-Ragoucy reflection algebras, which we call twisted super Yangians of type AIII, and classify their finite-dimensional irreducible representations under certain conditions. These superalgebras are coideal subalgebras of the super Yangian $\mathscr{Y}(\mathfrak{gl}_{m|n})$ and are associated with symmetric pairs of type AIII in Cartan's classification. We establish the Schur-Weyl type duality between degenerate affine Hecke algebras of type BC and twisted super Yangians.

Twisted super Yangians of type AIII and their representations

Abstract

We study the super analogue of the Molev-Ragoucy reflection algebras, which we call twisted super Yangians of type AIII, and classify their finite-dimensional irreducible representations under certain conditions. These superalgebras are coideal subalgebras of the super Yangian and are associated with symmetric pairs of type AIII in Cartan's classification. We establish the Schur-Weyl type duality between degenerate affine Hecke algebras of type BC and twisted super Yangians.
Paper Structure (28 sections, 53 theorems, 260 equations)

This paper contains 28 sections, 53 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Super Yangian
  3. General linear Lie superalgebras
  4. Super Yangians
  5. Highest weight representations
  6. Twisted super Yangian of type AIII
  7. Definition
  8. Basic properties of twisted super Yangian
  9. Highest weight representations
  10. Highest weight representations
  11. Verma modules
  12. Tensor product of representations
  13. Classifications in rank 1
  14. Non-super case
  15. Super case
  16. ...and 13 more sections

Key Result

Theorem 2.2

Given any total ordering on the elements $t_{ij}^{(p)}$ for $1\leqslant i,j\leqslant \varkappa$ and $p\in\mathbb Z_{>0}$, the ordered monomials in these elements, containing no second or higher order powers of the odd generators, form a basis of the super Yangian ${\mathscr{Y}_{\bm s}}$.

Theorems & Definitions (101)

  • Definition 2.1
  • Theorem 2.2: Gow2007gaussPeng2016parabolic
  • Definition 2.3
  • Theorem 2.4: Zhang1996super
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Proposition 2.7: Ragoucy2007analyticalBelliard2009nested
  • proof
  • Lemma 2.8
  • ...and 91 more