DG structures on categorified quantum groups

Acknowledgments
Introduction
Categorification
Quantum Groups
Odd Differentials
Motivations from link homology theory
The oddification program
Covering Kac-Moody algebras
Differential graded structures on categorified quantum group
Categorification
A DG-extension of Symmetric Functions
A DG-Structure on Odd 2-category
Super dg theory
Super 2-categories
Super dg-algebras
Super dg-categories
Super dg-2-categories
Hopfological algebra
Basic setup
DG-algebras from the Hopfological perspective
Decategorification from the Hopfological perspective
Gaussian integers
Results on Grothendieck groups of super dg-algebras
Grothendieck group of a super dg-algebra
Positively graded dg-algebras
Grothendieck ring of super dg-2-categories
Fantastic filtrations
The odd 2-category for sl(2)
The odd nilHecke ring
The odd categorified quantum group
Additional Relations
Derivations on the odd 2-category
Derivations on odd 2-category arising from the fantastic filtration
Differentials on the odd 2-category
Quantum sl(2) at a fourth root of unity
Idempotented quantum sl2
Small quantum sl2
Categorification results
Divided power modules
The DG-Grothendieck ring
A DG-extension of Symmetric Functions
The nilHecke algebra
The definition
The extended nilHecke algebra
The definition
Action on polynomials
Differentials
The ring of extended symmetric polynomials
Definition
The size of extended symmetric functions
Structure of the extended nilHecke algebra
Bases of extn
Combinatorial identities
Solomon's Theorem
Superpolynomials and superinvariants
Action of the extended nilHecke algebra
Preliminary computations
NHextn–equivariant isomorphisms
Example
Differentials
The standard differential
Koszul complex
Deformed differentials
Categorification
Conclusions and Future Work
Bibliography