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Gelfand-Tsetlin bases of representations for super Yangian and quantum affine superalgebra

Kang Lu

TL;DR

The paper develops explicit Gelfand-Tsetlin bases for skew representations of the super Yangian $Y(\mathfrak{gl}_{m|n})$ and derives exact actions of Drinfeld generators on these bases, proving irreducibility of skew representations. It introduces and leverages the Gelfand-Tsetlin algebra $\mathrm{A}(\mathfrak{gl}_{m|n})$ to define and analyze tame and thin modules, and furnishes a complete description for $Y(\mathfrak{gl}_{1|1})$ tameness. Extending the framework to the quantum affine superalgebra $U_q(\widehat{\mathfrak{gl}}_{m|n})$, the authors provide analogous GT-based Irreducibility results for skew representations, including evaluation morphisms, $\ell$-weights, and explicit action formulas. The work thereby unifies and advances understanding of GT bases, explicit current actions, and the pole structure in both the classical and quantum supersymmetric settings, with potential implications for tame module classifications and connections to related algebras.

Abstract

We give explicit actions of Drinfeld generators on Gelfand-Tsetlin bases of super Yangian modules associated with skew Young diagrams. In particular, we give another proof that these representations are irreducible. We study irreducible tame $\mathrm Y(\mathfrak{gl}_{1|1})$-modules and show that a finite-dimensional irreducible $\mathrm Y(\mathfrak{gl}_{1|1})$-module is tame if and only if it is thin. We also give the analogous statements for quantum affine superalgebra of type A.

Gelfand-Tsetlin bases of representations for super Yangian and quantum affine superalgebra

TL;DR

The paper develops explicit Gelfand-Tsetlin bases for skew representations of the super Yangian and derives exact actions of Drinfeld generators on these bases, proving irreducibility of skew representations. It introduces and leverages the Gelfand-Tsetlin algebra to define and analyze tame and thin modules, and furnishes a complete description for tameness. Extending the framework to the quantum affine superalgebra , the authors provide analogous GT-based Irreducibility results for skew representations, including evaluation morphisms, -weights, and explicit action formulas. The work thereby unifies and advances understanding of GT bases, explicit current actions, and the pole structure in both the classical and quantum supersymmetric settings, with potential implications for tame module classifications and connections to related algebras.

Abstract

We give explicit actions of Drinfeld generators on Gelfand-Tsetlin bases of super Yangian modules associated with skew Young diagrams. In particular, we give another proof that these representations are irreducible. We study irreducible tame -modules and show that a finite-dimensional irreducible -module is tame if and only if it is thin. We also give the analogous statements for quantum affine superalgebra of type A.

Paper Structure

This paper contains 18 sections, 28 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Super Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$
  3. Lie superalgebra $\mathfrak{gl}_{m|n}$
  4. Gelfand-Tsetlin tableaux
  5. Super Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$
  6. Gauss decomposition and $\ell$-weights
  7. Skew representations
  8. Main results for super Yangian
  9. Main results
  10. Proof of Theorem \ref{['thm:main']}
  11. Proof of Theorem \ref{['thm:irr']}
  12. Tame modules of $\mathrm{Y}(\mathfrak{gl}_{1|1})$
  13. Quantum affine superalgebra
  14. Quantum superalgebra $\mathrm{U}_q(\mathfrak{gl}_{m|n})$
  15. Quantum affine superalgebra $\mathrm{U}_q(\widehat{\mathfrak{gl}}_{m|n})$
  16. ...and 3 more sections

Key Result

Theorem 2.1

The $\mathfrak{gl}_{m|n}$-module $L(\lambda/\mu)$ admits a basis $\xi_{\Lambda}$ parameterized by all $\lambda/\mu$-admissible GT tableaux $\Lambda$. The actions of the generators of $\mathfrak{gl}_{m|n}$ are given by the formulas and for $m+1\leqslant k\leqslant m+n-1$, Here $\vartheta_{k,i}=\theta_{k1}+\ldots+\theta_{k,i-1}+\theta_{k-1,i+1}+\ldots+\theta_{k-1,m'}$. The arrays $\Lambda\pm \delt

Theorems & Definitions (54)

  • Conjecture 1.1
  • Theorem 2.1: SV10
  • Lemma 2.2: Gow05
  • Lemma 2.3: LM20
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 44 more