2017/03/17 by Zaur Guliyev, Guliyev, Zaur
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1703.05968
openalex publication_date 2017/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Khovanov-Lauda define a 2-category U such that the split Grothendieck group K0(U) is isomorphic to an integral version of the quantized universal enveloping algebra U(\mathfraksln), n ≥ 2. Beliakova-Habiro-Lauda-Webster prove that the trace decategorification of the Khovanov-Lauda 2-category is isomorphic to the the current algebra U(\mathfraksln [t]) - the universal enveloping algebra of the Lie algebra \mathfraksln ⊗ ℂ [t]. A 2-representation of U is a 2-functor from U to a linear, additive 2-category. In this note we are interested in the 2-representation, defined by Khovanov-Lauda using bimodules over cohomology rings of flag varieties. This 2-representation induces an action of the current algebra U(\mathfraksln [t]) on the cohomology rings. We explicitly compute the action of U(\mathfraksln [t]) generators using the trace functor. It turns out that the obtained current algebra module is related to another family of U(\mathfraksln [t])-modules, called local Weyl modules. Using known results about the cohomology rings, we are able to provide a new proof of the character formula for the local Weyl modules.