Representations of large Mackey Lie algebras and universal tensor categories

2023/09/05 by Penkov, Ivan, Tsanov, Valdemar
#16S37 #17B10 #17B65 #18E10 #18M05 #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2309.02522

Abstract

We extend previous work by constructing a universal abelian tensor category \bf T_t generated by two objects X,Y equipped with finite filtrations 0\subsetneq X₀\subsetneq ... X_t+1\subsetneq X and 0\subsetneq Y₀\subsetneq ... Y_t+1\subsetneq Y, and with a pairing X⊗ Y→ \mathbbI, where \mathbbI is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra \mathfrakgl^M(V,V_*) of cardinality 2^ℵ_t, associated to a diagonalizable pairing between two complex vector spaces V,V_* of dimension ℵ_t. As a preliminary step, we study a tensor category \mathbbT_t generated by the algebraic duals V^*, (V_*)^*. The injective hull of ℂ in \mathbbT_t is a commutative algebra I, and the category \bf T_t is consists of the free I-modules in \mathbbT_t. An essential novelty in our work is the explicit computation of Ext-groups between simples in both categories \bf T_t and \mathbbT_t, which had been an open problem already for t=0. This provides a direct link from the theory of universal tensor categories to Littlewood-Richardson-type combinatorics.

Related

• Universal tensor categories generated by dual pairs
• Representation categories of Mackey Lie algebras as universal monoidal\n categories
• Ordered tensor categories and representations of the Mackey Lie algebra of infinite matrices
• Three results on representations of Mackey Lie algebras
• Representation theory of Mackey Lie algebras and their dense subalgebras