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

2015/12/27 by Chirvasitu, Alexandru, Penkov, Ivan
#16T15 #17B10 #17B65 #Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1512.08157

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 \mathfrakgl^M(V,V_*). Here \mathfrakgl^M(V,V_*) is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces V_*⊗ V→\mathbbK, where \mathbbK is the base field. Tensor representations of \mathfrakgl^M(V,V_*) 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 \mathbbT³_\mathfrakgl^M(V,V_*) which they comprise, extends a category \mathbbT_\mathfrakgl^M(V,V_*) previously studied in [4, 12,17], and our main result is that \mathbbT³_\mathfrakgl^M(V,V_*) 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 \mathbbT_\mathfrakgl^M(V,V_*) established in [12]. Finally, we discuss the extension of \mathbbT³_\mathfrakgl^M(V,V_*) by the algebraic dual (V_*)^* of V_*.

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• Representations of large Mackey Lie algebras and universal tensor categories
• Representation theory of Mackey Lie algebras and their dense subalgebras
• Three results on representations of Mackey Lie algebras
• Universal tensor categories generated by dual pairs
• Topological tensor representations of gl(V) for a space V of countable dimension