Representations of large Mackey Lie algebras and universal tensor categories

Abstract

We extend previous work by constructing a universal abelian tensor category ${\mathbf{T}}_t$ generated by two objects $X,Y$ equipped with finite filtrations $0⊊X_0⊊...X_{t+1}⊊X$ and $0⊊Y_0⊊...Y_{t+1}⊊Y$, 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 ${\mathfrak{gl}}^M(V,V_{\ast })$ of cardinality $2^{{\aleph }_t}$, associated to a diagonalizable pairing between two complex vector spaces $V,V_{\ast }$ of dimension ${\aleph }_t$. As a preliminary step, we study a tensor category ${\mathbb{T}}_t$ generated by the algebraic duals $V^{\ast }$, $(V_{\ast }{)}^{\ast }$. The injective hull of $\mathbb{C}$ in ${\mathbb{T}}_t$ is a commutative algebra $I$, and the category ${\mathbf{T}}_t$ 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 ${\mathbf{T}}_t$ 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.

41 pages