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.

Quantum supergroups VI: roots of 1

“…We expect that the quantum covering groups of finite type at roots of 1 have very interesting representation theory, which has yet to be developed (compare [AJS94]). The categorification of the quantum covering group of rank one at roots of 1 is already highly nontrivial as shown in the recent work of Egilmez and Lauda [EgL18]. We hope our work on higher rank quantum covering groups could provide a solid algebraic foundation for further super categorification and connection to quantum topology.…”

𝑝-DG Cyclotomic nilHecke Algebras II

“…In another parallel direction, the odd version of the nilHecke algebra (defined over ) was equipped with a DG structure in [EQ16c] where it was shown that the Grothendieck group is isomorphic to the positive half of quantum sl 2 at a fourth root of unity. This DG structure was extended to the entire odd sl 2 category in [EL18], which gives rise to a categorification of the entire quantum group for sl 2 at a fourth root of unity, as well as a certain subalgebra of gl(1|1). These structures should give rise to an algebraic explanation of the close relationship between special values of Alexander polynomial and the Jones polynomial.…”

Not even Khovanov homology

“…After the appearance of odd Khovanov homology in [15] there has been a certain interest in odd categorified structures and supercategorification (see for example [2,3,4,5,6,7,11,14]). In contrast to (even) Khovanov homology, odd Khovanov homology has an anticommutative feature.…”