Representation Theory of the Twisted Yangians in Complex Rank

TL;DR

This paper extends the theory of Yangians in complex rank by defining twisted Yangians in Deligne categories and classifying their simple modules, generalizing previous work on untwisted Yangians.

Contribution

It introduces the notion of twisted Yangians in Deligne categories and classifies their finite-length simple modules, expanding the understanding of these algebraic structures in complex rank.

Findings

Defined twisted Yangians in Deligne categories

Classified simple modules over $Y(rak{o}_t)$ and $Y(rak{sp}_t)$

Extended techniques from previous Yangian classifications

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.