Seminar

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Ben Webster defined a family of finite-dimensional algebras, now called KLRW algebras, whose module categories categorify tensor products of irreducible representations of the quantum group of a semisimple Lie algebra. Moreover, the set of simple modules carries a natural crystal structure which is isomorphic to the underlying crystal structure of the decategorified representation. Work of André Henriques and Joel Kamnitzer shows the cactus group has an action on tensor products of crystals; via the isomorphism, the set of simple modules over a KLRW algebra inherits an action of the cactus group. This raises the question of whether the cactus action can be defined intrinsically from the category of KLRW modules. We expect the answer to come from taking an appropriate crystal limit of the braid group action on the derived category of modules. This talk will discuss current progress on this question.