Crystals and coboundary categories

By A. Henriques, Joel Kamnitzer. 2004

119 citations recorded by the source corpus.

Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category. Similar to the case of braided categories, there is a group naturally acting on multiple tensor products in coboundary categories. We call this group the cactus group and identify it as the fundamental group of the moduli space of marked real genus zero stable curves.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/011ac9e31659cee36021dd422d39eb78bdc00011

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In Algebra and Number Theory

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Citations in the corpus

Continuous crystal and Duistermaat-Heckman measure for Coxeter groups (2008, methodology): “[16] A. Henriques and J. Kamnitzer, Crystals and coboundary categories.”
The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points (2005, background): “Let C be such a category, and let X1, . . . , Xn be n objects in C. Then, as shown in [ HK ], every element g of the group Jn defines a morphism X1 ⊗ � � � ⊗ Xn → Xg(1) ⊗ � � � ⊗ Xg(n) (here g(i) is the action of the image of g in Sn on the index i). Namely, sp,q acts by”
The octahedron recurrence and gln crystals (2004, background): “in a way compatible with the associator and the commutor. Then any axiom satisﬁed in C(such as the pentagon axiom used in the deﬁnition of monoidal categories) will automatically be satisﬁed in D. In [HK1], we proved that Crystalsis a coboundary category. Namely, it is a monoidal category equipped w”
ON THE COMBINATORICS OF CRYSTAL GRAPHS, I. LUSZTIG'S INVOLUTION (2005, unknown): “The importance of this involution stems from the fact that it exhibits the crystals as self-dual posets, and it corresponds to the action of the longest Weyl group element on an irreducible representation; it also appears in other contexts, such as the recent realization of the category of crystals ”