Let g denote a semisimple Lie algebra. Lusztig introduced an action of the braid group on any
integrable representation of the quantum group of g. This action was realized
categorically in work of Chuang-Rouquier, as shown by Cautis-Kamnitzer, where
each braid group generator is upgraded to a complex of functors called a Rickard
complex. I will describe a corresponding action of the cactus group on a g-crystal
coming from a representation. In joint work with Licata, Losev and Yacobi, we
show that this action can be recovered categorically from the Rickard complexes,
when considering the positive lifts of the longest Weyl group elements for certain
parabolics in g.

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