Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra g has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given g, there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When g is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of g, and its edges are parallel to the roots of g. In this paper, we generalize this construction to the case where g is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as g. The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott's tilting theory for the category Λ-mod. The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.One of these tools is Buan, Iyama, Reiten and Scott's tilting ideals for Λ [14]. Let S i be the simple Λ-module of dimension-vector α i and let I i be its annihilator, a one-codimensional two-sided ideal of Λ. The products of these ideals I i are known to satisfy the braid relations, so to each w in the Weyl group of g, we can attach a two-sided ideal I w of Λ by the rule I w = I i 1 · · · I i ℓ , where s i 1 · · · s i ℓ is any reduced decomposition of w. Given a finite-dimensional Λ-module T , we denote the image of the evaluation map I w ⊗ Λ Hom Λ (I w , T ) → T by T w .Recall that the dominant Weyl chamber C 0 and the Tits cone C T are the convex cones in the dual of RI defined asWe will show the equality T min θ = T max θ = T w for any finite dimensional Λ-module T , any w ∈ W and any linear form θ ∈ wC 0 . This implies that dim T w is a vertex of Pol(T ) and that the normal cone to Pol(T ) at this vertex contains wC 0 . This also implies that Pol(T ) is contained inWhen θ runs over the Tits cone, it generically belongs to a chamber, and we have just seen that in this case, the face P θ is a vertex. When θ lies on a facet, P θ is an edge (possibly degenerate). More precisely, if θ lies on the facet that separates the chambers wC 0 and ws i C 0 , with say ℓ(ws i ) > ℓ(w), then (T min θ , T max θ ) = (T ws i , T w ). Results in [1] and [27] moreover assert that T w /T ws i is the direct sum of a finite number of copies of the Λ-module I w ⊗ Λ S i .There is a similar description when θ is in −C T ; here the submodules T w of T that come into play are the kernels of the coevaluation maps T → ...
It has previously been shown that, at least for non-exceptional Kac-Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals. In particular, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to Macdonald polynomials and q-deformed Whittaker functions.
Abstract. We describe how Mirković-Vilonen polytopes arise naturally from the categorification of Lie algebras using Khovanov-Lauda-Rouquier algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes.MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for all symmetrizable Kac-Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate the polytopes with some extra information. We suggest that the resulting "KLR polytopes" are the general-type analogues of MV polytopes.We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Baumann, Kamnitzer and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.
Abstract. Drinfeld defined a unitarized R-matrix for any quantum group Uq(g). This gives a commutor for the category of Uq(g) representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives Uq(g) representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer's construction agrees with Drinfeld's commutor. We then describe the action of Drinfeld's commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor.
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