Representations of large Mackey Lie algebras and universal tensor categories

Abstract

We extend previous work by constructing a universal abelian tensor category \(\textbf{T}_t\) generated by two objects X, Y equipped with finite filtrations \(0\subsetneq X_0\subsetneq ...\subsetneq X_{t+1}= X\) and \(0\subsetneq Y_0\subsetneq ... \subsetneq Y_{t+1}= Y\), and with a pairing \(X\otimes Y\rightarrow \mathbbm {1}\), where \(\mathbbm {1}\) is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra \(\mathfrak {gl}^M(V,V_*)\) of cardinality \(2^{\aleph _t}\), associated to a diagonalizable pairing between two vector spaces \(V,V_*\) of dimension \(\aleph _t\) over an algebraically closed field \({{\mathbb {K}}}\) of characteristic 0. As a preliminary step, we study a tensor category \({{\mathbb {T}}}_t\) generated by the algebraic duals \(V^*\) and \((V_*)^*\). The injective hull of the trivial module \({{\mathbb {K}}}\) in \({{\mathbb {T}}}_t\) is a commutative algebra I, and the category \(\textbf{T}_t\) consists of all free I-modules in \({{\mathbb {T}}}_t\). An essential novelty in our work is the explicit computation of Ext-spaces between simples in both categories \(\textbf{T}_t\) and \({{\mathbb {T}}}_t\), 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.