Towards interpolating categories for equivariant map algebras

2025/04/29 by Saima Samchuck-Schnarch, Samchuck-Schnarch, Saima
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2504.21163

Abstract

Using the language of string diagrams, we define categorical generalizations of modules for map algebras \mathfrakg ⊗ A and equivariant map algebras (\mathfrakg ⊗ A)^Γ, where \mathfrakg is a Lie algebra, A is a commutative associative algebra, and Γ is an abelian group acting on \mathfrakg and A. After establishing some properties of these modules, we present several examples of how our definitions can applied in various diagrammatic categories. In particular, we use the oriented Brauer category OB to construct a candidate interpolating category for the categories of \mathfrakgl_n ⊗ k[t]-modules.

Citations

• Diagrammatics for real supergroups
• Representations of the oriented skein category
• Lie monads and dualities
• The Brauer Category and Invariant Theory
• Irreducible finite-dimensional representations of equivariant map algebras
• A categorical approach to Weyl modules
• Representations of Multiloop Algebras
• Multi-dimensional Weyl Modules and Symmetric Functions
• Spiders for rank 2 Lie algebras

Related

• Irreducible finite-dimensional representations of equivariant map algebras
• Diagrammatic Categories which arise from Representation Graphs
• Modules over the category of Jacobi diagrams in handlebodies
• Extensions des modules de dimension finie pour les algèbres de courants tordues
• On G--equivariant modular categories