Representations of the cyclotomic oriented Brauer-Clifford supercategory

Abstract

Let $k$ be an algebraically closed field with characteristic p different from 2. We generalize the notion of a weakly triangular decomposition in [7] to the super case called a super weakly triangular decomposition. We show that the underlying even category of locally finite-dimensional left A-supermodules is an upper finite fully stratified category in the sense of [6, Definition 3.34] if the superalgebra A admits an upper finite super weakly triangular decomposition. In particular, when A is the locally unital superalgebra associated with the cyclotomic oriented Brauer-Clifford supercategory in [1], the Grothendieck group of the category of left A-supermodules admitting finite standard flags has a $g$-module structure that is isomorphic to the tensor product of an integrable lowest weight and an integrable highest weight $g$-module, where $g$ is the complex Kac-Moody Lie algebra of type $A_{2\ell }^{\left(2\right)}$ (resp., $B_{\infty }$) if $p=2\ell +1$ (resp., $p=0$).