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Representations of the modular shifted super Yangian $Y_{1|1}(σ)$

Hao Chang, Ruiying Hou, Hui Wu

Abstract

Let $Y_{1|1}$ be the Yangian associated to the general linear Lie superalgebra $\mathfrak{gl}_{1|1}$, defined over an algebraically closed field $\mathbbm{k}$ of characteristic $p>2$. In this paper, we classify the finite dimensional irreducible representations of the restricted super Yangian $Y_{1|1}^{[p]}$ and the restricted truncated shifted super Yangian $Y_{1|1,\ell}^{[p]}(σ)$.

Representations of the modular shifted super Yangian $Y_{1|1}(σ)$

Abstract

Let be the Yangian associated to the general linear Lie superalgebra , defined over an algebraically closed field of characteristic . In this paper, we classify the finite dimensional irreducible representations of the restricted super Yangian and the restricted truncated shifted super Yangian .
Paper Structure (14 sections, 27 theorems, 125 equations)

This paper contains 14 sections, 27 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Restricted Yangians
  3. Modular Yangian $Y_{1|1}$
  4. Restricted super Yangian $Y_{1|1}^{[p]}$
  5. Hopf algebra structure
  6. Evaluation homomorphism
  7. Representations of ${\rm Y}^{[p]}$
  8. Baby Verma modules
  9. Finite dimensionality conditions
  10. Restricted shifted super Yangians
  11. Shifted super Yangains
  12. Irreducible representations of $\mathop{\mathrm{Y}}\nolimits_{\sigma,\ell}^{[p]}$
  13. Proof of Proposition \ref{['prop: delta send p to p']}
  14. Hopf algebra structure.

Key Result

Theorem 2.1

The super Yangian $\mathop{\mathrm{Y}}\nolimits$ is generated by the elements $\{d_i^{(r)}, d_i^{\prime (r)}; 1 \leq i \leq 2, r>0\}$ and $\{e^{(r)},f^{(r)}; r>0\}$ subject only to the following relations:

Theorems & Definitions (55)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 45 more