Representations of Brauer category and categorification

Abstract

We study representations of the locally unital and locally finite dimensional algebra B associated to the Brauer category B(δ0) with defining parameter δ0 over an algebraically closed field K with characteristic p≠2. The Grothendieck group K0(B-modΔ) will be used to categorify the integrable highest weight slK-module V(ϖδ0−12) with the fundamental weight ϖδ0−12 as its highest weight, where B-modΔ is a subcategory of B-lfdmod in which each object has a finite Δ-flag, and slK is either sl∞ or slˆp depending on whether p=0 or 2∤p. As g-modules, C⊗ZK0(B-modΔ) is isomorphic to V(ϖδ0−12), where g is a Lie subalgebra of slK (see Definition 4.2). When p=0, standard B-modules and projective covers of simple B-modules correspond to monomial basis and so-called quasi-canonical basis of V(ϖδ0−12), respectively.