DG structures on odd categorified quantum $sl(2)$

Abstract

We equip Ellis and Brundan’s version of the odd categorified quantum group for $sl(2)$ with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading gives rise to two possible decategorifications of the associated dg-2-category. One version gives rise to a categorification of quantum $sl(2)$ at a fourth root of unity, while the other version produces a subalgebra of quantum $gl(1|1)$ defined over the integers. Both of these algebras appear in connection with quantum algebraic approaches to the Alexander polynomial.