Categorical Quantum Groups and Braided Monoidal 2Categories
Abstract
Following the dimensional ladder, we develop a systematic categorification of the theory of quantum groups/bialgebras in the $A_\infty$ setting, and study their higherrepresentation theory. By following closely the generalized quantum double construction of Majid, we construct in particular the 2quantum double $D(\mathcal{G})$ associated to a 2bialgebra $\mathcal{G}$, and prove its duality and factorization properties. We also characterize a notion of (quasitriangular) 2Rmatrix $\mathcal{R}$ and identify the associated 2YangBaxter equations, which can be seen as a categorification of the usual notion of $R$matrix in an ordinary quantum group. The main result we prove in this paper is that the weak 2representation 2category $\operatorname{2Rep}^{\mathcal{T}}(\mathcal{G})$ of a quasitriangular 2bialgebra $(\mathcal{G},\mathcal{T},\mathcal{R})$  when monoidally weakened by a Hochschild 3cocycle $\mathcal{T}$  forms a braided monoidal 2category.
 Publication:

arXiv eprints
 Pub Date:
 April 2023
 DOI:
 10.48550/arXiv.2304.07398
 arXiv:
 arXiv:2304.07398
 Bibcode:
 2023arXiv230407398C
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Category Theory;
 16T10 (primary);
 18M15 (secondary);
 18N70
 EPrint:
 51 pages, 1 figure (v2: clarified certain parts