Trace and categorical sl(n) representations

Abstract

define a 2-category U such that the split Grothendieck group K 0 (U ) is isomorphic to an integral version of the quantized universal enveloping algebra U(sl n ), n ≥ 2. Beliakova-Habiro-Lauda-Webster [2] prove that the trace decategorification Tr(U ) of the Khovanov-Lauda 2-category is isomorphic to the the current algebra U(sl n [t]) -the universal enveloping algebra of the Lie algebra sl n ⊗ 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(sl n [t]) on the cohomology rings. We explicitly compute the action of U(sl n [t]) generators using the trace functor. It turns out that the obtained current algebra module is related to another family of U(sl n [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.