Representations of large Mackey Lie algebras and universal tensor categories

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