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 $\mathcal{B}({\delta }_0)$ with defining parameter ${\delta }_0$ over an algebraically closed field $K$ with characteristic $p\ne 2$. The Grothendieck group $K_0(B{\text{-mod}}^{\Delta })$ will be used to categorify the integrable highest weight ${\mathfrak{sl}}_K$-module $V({\varpi }_{\frac{{\delta }_0-1}{2}})$ with the fundamental weight ${\varpi }_{\frac{{\delta }_0-1}{2}}$ as its highest weight, where $B$-mod${}^{\Delta }$ is a subcategory of $B$-lfdmod in which each object has a finite $\Delta$-flag, and ${\mathfrak{sl}}_K$ is either ${\mathfrak{sl}}_{\infty }$ or ${\widehat{\mathfrak{sl}}}_p$ depending on whether $p=0$ or $2\nmid p$. As $\mathfrak{g}$-modules, $\mathbb{C}{\otimes }_{\mathbb{Z}}{K}_0(B{\text{-mod}}^{\Delta })$ is isomorphic to $V({\varpi }_{\frac{{\delta }_0-1}{2}})$, where $\mathfrak{g}$ is a Lie subalgebra of ${\mathfrak{sl}}_K$ (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 $V({\varpi }_{\frac{{\delta }_0-1}{2}})$, respectively.

23 pages