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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∗)⊗m ⊗ (V∗)⊗n ⊗ V⊗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 Dan-Cohen et al. Adv. Math. 289, 205–278, (2016), Penkov and Serganova (2014) and Sam and Snowden Forum Math. Sigma 3(e11):108, (2015) . 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 Penkov and Serganova (2014). Finally, we discuss the extension of \(\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}\) obtained by adjoining the algebraic dual (V∗)∗ of V∗.

Ordered Tensor Categories and Representations of the Mackey Lie Algebra of Infinite Matrices