DG structures on odd categorified quantum sl(2)

TL;DR

This paper introduces a differential structure on an odd categorified quantum sl(2), leading to two decategorifications that relate to quantum sl(2) at roots of unity and quantum gl(1|1), with applications to the Alexander polynomial.

Contribution

It provides a new differential graded supercategory structure on Ellis and Brundan's odd categorified quantum sl(2), revealing two distinct decategorifications linked to quantum algebra.

Findings

Decategorification yields quantum sl(2) at a fourth root of unity.

Decategorification produces a subalgebra of quantum gl(1|1).

Connections to quantum algebraic approaches to the Alexander polynomial.

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.