Current algebras and categorified quantum groups

Abstract

We identify the trace, or 0th Hochschild homology, of type ADE categorified quantum groups with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of KLR-algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups.

28 pages, with tikz and xypic diagrams. v2: changed discussion of cyclicity to include all simply-laced Kac-Moody algebras. v3: revised final version. edited from journal version to fix some when ADE hypothesis is needed

References in corpus (16)

• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Knot invariants and higher representation theory
• Derived equivalences for symmetric groups and sl\_2-categorification
• On the center of quiver-Hecke algebras
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• Trace decategorification of categorified quantum sl(2)
• Trace as an alternative decategorification functor
• Rigidity in higher representation theory
• sl(3)-Foams and the Khovanov-Lauda categorification of quantum sl(k)
• sl(N)-Web categories
• Cyclicity for categorified quantum groups
• Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties
• Unfurling Khovanov-Lauda-Rouquier algebras
• On the definition of Kac-Moody 2-category
• Degeneration of trigonometric dynamical difference equations for quantum loop algebras to trigonometric Casimir equations for Yangians

Cited by in corpus (14)

• The cohomological Hall algebra of a preprojective algebra
• Trace decategorification of the Hecke category
• The Elliptic Hall algebra and the deformed Khovanov Heisenberg category
• Cyclicity for categorified quantum groups
• A graphical calculus for the Jack inner product on symmetric functions
• Weyl modules for Lie superalgebras
• Centers of KLR algebras and cohomology rings of quiver varieties
• Trace of the twisted Heisenberg Category
• Planar diagrammatics of self-adjoint functors and recognizable tree series
• Self-dual Grassmannian, Wronski map, and representations of ${\mathfrak{gl}}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$
• An equivalence between truncations of categorified quantum groups and Heisenberg categories
• Local Weyl modules and fusion products for the current superalgebra $\mathfrak{sl}(1|2)[t]$
• Frobenius W-algebras and traces of Frobenius Heisenberg categories
• Trace and categorical sl(n) representations