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 $Y({\mathfrak{gl}}_t)$ 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.