2-categorical affine symmetries of quantum enveloping algebras

Abstract

We produce 2-representations of the positive part of affine quantum enveloping algebras on their finite-dimensional counterparts in type $A_n$. These 2-representations naturally extend the right-multiplication 2-representation of $U_q^{+}({\mathfrak{sl}}_{n+1})$ on itself and are closely related to evaluation morphisms of quantum groups. We expect that our 2-representation exists in all simple types and show that the corresponding 1-representation exists in types $D_4$ and $C_2$. We also show that a certain quotient of our 1-representation in type $A_n$ is isomorphic to a prefundamental representation. We use this to provide a new proof of the prefundamental representation character formulas in these cases.

v4: Some arguments are simplified by relating our functor E_0 to the standard module for the highest root. A few expositional simplifications. References added. Minor spelling/style changes. 41 pages with tikz figures. Final version, to appear in Journal of Algebra