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Algebras and Representation Theory

, Volume 22, Issue 1, pp 249–279 | Cite as

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

  • Alexandru ChirvasituEmail author
  • Ivan Penkov
Article

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)m ⊗ (V)nVp 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.

Keywords

Mackey lie algebra Tensor category Koszulity 

Mathematics Subject Classification (2010)

17B65 17B10 16T15 

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Notes

Acknowledgments

We thank Vera Serganova for sharp comments on the topic of this paper, as well as the anonymous referee for a very careful reading and illuminating remarks on an initial draft.

A. C. was partially funded through NSF grant DMS-1565226. I. P. acknowledges continued partial support by the DFG through the Priority Program “Representation Theory” and through grant PE 980/6-1.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity at BuffaloBuffaloUSA
  2. 2.Jacobs University BremenBremenGermany

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