Representations of large Mackey Lie algebras and universal tensor categories

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 $0\subsetneq X_0\subsetneq ... X_{t+1}\subsetneq X$ and $0\subsetneq Y_0\subsetneq ... Y_{t+1}\subsetneq Y$, and with a pairing $X\otimes Y\to \mathbb{I}$, where $\mathbb{I}$ 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 complex vector spaces $V,V_*$ of dimension $\aleph_t$. As a preliminary step, we study a tensor category $\mathbb{T}_t$ generated by the algebraic duals $V^*$, $(V_*)^*$. The injective hull of $\mathbb{C}$ in $\mathbb{T}_t$ is a commutative algebra $I$, and the category ${\bf T}_t$ is consists of the free $I$-modules in $\mathbb{T}_t$. An essential novelty in our work is the explicit computation of Ext-groups between simples in both categories ${\bf T}_t$ 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.