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

TL;DR

This paper introduces ordered tensor categories related to the representation theory of the Mackey Lie algebra of infinite matrices, establishing their structure as finite-length, Koszul, self-dual tensor categories with universal properties.

Contribution

It defines and analyzes a new tensor category for representations of the Mackey Lie algebra, proving its finite length, Koszulity, self-duality, and universal property, extending previous work.

Findings

The category is finite-length and Koszul.

It is self-dual and has a universal property.

Simplifies proofs of earlier results.

Abstract

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 $V_*$.