Research Project: Our research interests are connected with representation theory of quantum groups. For instance, recently, we jointly introduced and studied actions of various groups generated by involutions on categories representations of the quantized enveloping algebra U_q(g) corresponding to a semisimple Lie algebra g. Those actions generalize Kashiwara and others' symmetries on
crystal bases. The groups in question first appeared in algebraic geometry and have numerous applications in topology and mathematical physics.

One of our present projects aims to geometrize these actions using unipotent crystals for the Langlands dual of the Lie group of g. These would result in constructing the "tropical mirror" of representations of the quantum group U_q(g).

Our other project is to construct new braid group actions and their generalizations on various categories by introducing and solving a generalization of the quantum Yang-Baxter equation as well as its classical counterparts. This was motivated by non-standard braidings on representations of U_q(g) obtained previously in our study of quantum folding. 

Research Team: Jianrong Li (U of Vienna), Arkady Berenstein (U of Oregon),  Jacob Greenstein (U of California, Riverside)