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Type C blocks of super category \(\mathcal{O}\). (English) Zbl 1446.17029
Authors’ abstract: “We show that the blocks of category \(\mathcal O\) for the Lie superalgebra \(\mathfrak q_n(\mathbb C)\) associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type \(C_\infty\). This allows us to prove two conjectures formulated by Cheng, Kwon and Wang. We then focus on the full subcategory consisting of finite-dimensional representations, which we show is a highest weight category with blocks that are Morita equivalent to certain generalized Khovanov arc algebras.”
The Lie superalgebra \(\mathfrak q_n(\mathbb C)\) is the subalgebra of the general linear Lie superalgebra \(\mathfrak{gl}_{n|n}(\mathbb C)\), consisting of matrices of the form \(\begin{pmatrix} A & B \\ B & A \end{pmatrix}\). The main ingredient is some involved combinatorics related to tensor product modules over the Lie algebras \(\mathfrak{sl}_{+\infty}\) and \(\mathfrak{sp}_{2\infty}\).

MSC:
17B45 Lie algebras of linear algebraic groups
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Citations:
Zbl 1406.17014
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