DG structures on categorified quantum groups

2018-07-16This thesis studies DG structures on categorified quantum groups. In the first part of the thesis, we equip Ellis and Brundan's version of the odd categorified quantum group for sl₂ with a differential and identify the Grothendieck ring of the associated dg-2-category with a categorification of sl₂ at a fourth root of unity. In the second part of the thesis we investigate analogs of symmetric functions arising from an extension of the nilHecke algebra defined by Naisse and Vaz. These extended symmetric functions form a subalgebra of the polynomial ring tensored with an exterior algebra. We define families of bases for this algebra and show that it admits a family of differentials making it a sub-DG-algebra of the extended nilHecke algebra. The ring of extended symmetric functions equipped with this differential is quasi-isomorphic to the cohomology of a Grassmannian. We also introduce new deformed differentials on the extended nilHecke algebra that when restricted makes extended symmetric functions quasi-isomorphic to GL(N)-equivariant cohomology of Grassmannians