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

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 $V_*$.