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We equip Ellis and Brundan’s version of the odd categorified quantum group for 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 at a fourth root of unity, while the other version produces a subalgebra of quantum defined over the integers. Both of these algebras appear in connection with quantum algebraic approaches to the Alexander polynomial.
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Ilknur Egilmez, Aaron D. Lauda, DG structures on odd categorified quantum . Quantum Topol. 11 (2020), no. 2, pp. 227–294DOI 10.4171/QT/135