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A representation of\mathfrak{gl}(V)=V \otimes V^{\ast}is a linear map\mu \colon \mathfrak{gl}(V) \otimes M \rightarrow Msatisfying a certain identity. By currying, giving a linear map\muis equivalent to giving a linear mapa \colon V \otimes M \rightarrow V \otimes M, and one can translate the condition for\muto be a representation into a condition ona. This alternate formulation does not use the dual ofVand makes sense for any objectVin a tensor category\mathcal{C}. We call such objects representations of thecurried general linear algebraonV. The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category is the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.
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This content will become publicly available on March 20, 2027
Type II quantum subgroups of quantum $\mathfrak{sl}_{N}$. II: Classification
In this paper, we study the indecomposable module categories over\mathcal{C}(\mathfrak{sl}_{N}, k), the category of integrable level-krepresentations of affine Kac–Moody\mathfrak{sl}_{N}. Our first main result classifies these module categories in the case of generick; i.e.,kis sufficiently large relative toN. As\mathcal{C}(\mathfrak{sl}_{N}, k)is a braided tensor category, there is a relative tensor product structure on its category of module categories. In the generic setting, we obtain a formula for the relative tensor product rules between the indecomposable module categories. Our second main result classifies the indecomposable module categories over\mathcal{C}(\mathfrak{sl}_{N}, k)forN\leq 7, with no restrictions onk. In this non-generic setting, exceptional module categories are obtained. This work relies heavily on previous results by the two authors. In previous literature, module category classification results were known only for\mathfrak{sl}_{2}and\mathfrak{sl}_{3}.
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- Award ID(s):
- 1928930
- PAR ID:
- 10695511
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Quantum Topology
- ISSN:
- 1663-487X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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