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.