Crystals and coboundary categories

   Joel Kamnitzer, (UC Berkeley)

Abstract: A crystal for a representation of a semisimple Lie algebra is a combinatorial object which encodes the structure of the representation. There is an interesting tensor product on these crystals. We give a construction of a commutor (natural isomophisms A x B -> B x A) for the category of crystals of a semisimple Lie algebra. This commutor is symmetric but does not satisfy the usual hexagon axiom. Instead it obeys a different axiom which makes the category of crystals into a coboundary category. Motivated by the above construction, we investigate the structure of coboundary categories. Just as the braid group acts on repeated tensor products in a braided category, the fundamental group of the moduli space of stable real genus 0 curves with n marked points acts on repeated tensor products in a coboundary category.

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