Year: 2024
Type: article
Abstract: We extend previous work by constructing a universal abelian tensor category $$\textbf{T}_t$$ 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 ⊗ Y → 1 , where $$\mathbbm {1}$$ 1 is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra $$\mathfrak {gl}^M(V,V_*)$$ gl M ( V , V ∗ ) of cardinality $$2^{\aleph _t}$$ 2 ℵ t , associated to a diagonalizable pairing between two vector spaces $$V,V_*$$ V , V ∗ of dimension $$\aleph _t$$ ℵ t over an algebraically closed field $${{\mathbb {K}}}$$ K of characteristic 0. As a preliminary step, we study a tensor category $${{\mathbb {T}}}_t$$ T t generated by the algebraic duals $$V^*$$ V ∗ and $$(V_*)^*$$ ( V ∗ ) ∗ . The injective hull of the trivial module $${{\mathbb {K}}}$$ K in $${{\mathbb {T}}}_t$$ T t is a commutative algebra I, and the category $$\textbf{T}_t$$ T t consists of all free I-modules in $${{\mathbb {T}}}_t$$ T t . An essential novelty in our work is the explicit computation of Ext-spaces between simples in both categories $$\textbf{T}_t$$
Authors: Ivan Penkov, Valdemar Tsanov
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Award: G75940874
OpenAlex ID: W4402534235