Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representation Theory of the Twisted Yangians in Complex Rank

Arun S. Kannan, Shihan Kanungo

Abstract

In 2016, Etingof defined the notion of a Yangian in a symmetric tensor category and posed the problem to study them in the context of Deligne categories. This problem was studied by Kalinov in 2020 for the Yangian $Y(\mathfrak{gl}_t)$ of the general linear Lie algebra $\mathfrak{gl}_t$ in complex rank using the techniques of ultraproducts. In particular, Kalinov classified the simple finite-length modules over $Y(\mathfrak{gl}_t)$. In this paper, we define the notion of a twisted Yangian in Deligne's categories, and we extend these techniques to classify finite-length simple modules over the twisted Yangians $Y(\mathfrak{o}_t)$ and $Y(\mathfrak{sp}_t)$ of the orthogonal and symplectic Lie algebras $\mathfrak{o}_t,\mathfrak{sp}_t$ in complex rank.

Representation Theory of the Twisted Yangians in Complex Rank

Abstract

In 2016, Etingof defined the notion of a Yangian in a symmetric tensor category and posed the problem to study them in the context of Deligne categories. This problem was studied by Kalinov in 2020 for the Yangian of the general linear Lie algebra in complex rank using the techniques of ultraproducts. In particular, Kalinov classified the simple finite-length modules over . In this paper, we define the notion of a twisted Yangian in Deligne's categories, and we extend these techniques to classify finite-length simple modules over the twisted Yangians and of the orthogonal and symplectic Lie algebras in complex rank.
Paper Structure (17 sections, 51 theorems, 214 equations)

This paper contains 17 sections, 51 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. The representation categories RepGLN, RepSON, Rep(SpN)
  3. Representation categories of classical groups
  4. Representations of classical Lie algebras
  5. Representation categories
  6. Twisted Yangians
  7. The (twisted) Yangians Y(gl2n), Y(sp2n), Y(o2n)
  8. Representations of Y(gn)
  9. Finite-dimensional irreducible representations of $Y(\mathfrak{sp}_{2n})$ in positive characteristic
  10. Finite-dimensional irreducible representations of $Y(\mathfrak{o}_{2n})$ in positive characteristic
  11. Deligne's categories for classical groups
  12. The Deligne categories RepOt and RepSpt
  13. Ultraproduct construction
  14. Twisted Yangians in complex rank
  15. The twisted Yangian Y(g)
  16. ...and 2 more sections

Key Result

Proposition 2.2

For any $\lambda,\mu \in C_{\mathbb{Z}} \cap X(H)^+$, we have $\mathcal{L}(\lambda) = \Delta(\lambda)$, and moreover there are no non-trivial extensions between $\mathcal{L}(\lambda)$ and $\mathcal{L}(\mu)$.

Theorems & Definitions (96)

  • Remark 2.1
  • Proposition 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 86 more