Ordered tensor categories and representations of the Mackey Lie algebra of infinite matrices

Alexandru Chirvasitu, Ivan Penkov

Published 2015-12-27Version 1

We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices $\mathfrak{gl}^M\left(V,V_*\right)$. Here $\mathfrak{gl}^M\left(V,V_*\right)$ is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces $V_*\otimes V\to\mathbb{K}$, where $\mathbb{K}$ is the base field. Tensor representations of $\mathfrak{gl}^M\left(V,V_*\right)$ are defined as arbitrary subquotients of finite direct sums of tensor products $(V^*)^{\otimes m}\otimes (V_*)^{\otimes n}\otimes V^{\otimes p}$ where $V^*$ denotes the algebraic dual of $V$. The category $\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ which they comprise, extends a category $\mathbb{T}_{\mathfrak{gl}^M\left(V,V_*\right)}$ previously studied in [4, 12,17], and our main result is that $\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ is a finite-length, Koszul self-dual, tensor category with a certain universal property that makes it into a ``categorified algebra'' defined by means of a handful of generators and relations. This result uses essentially the general properties of ordered Grothendieck categories, which yield also simpler proofs of some facts about the category $\mathbb{T}_{\mathfrak{gl}^M\left(V,V_*\right)}$ established in [12]. Finally, we discuss the extension of $\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ by the algebraic dual $(V_*)^*$ of $V_*$.