Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given , there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When 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 , and its edges are parallel to the roots of . In this paper, we generalize this construction to the case where 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 2-faces. 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 . The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category . The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.
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DOI: 10.1007/s10240-013-0057-y
Pierre Baumann 1; Joel Kamnitzer 2; Peter Tingley 3
@article{PMIHES_2014__120__113_0,
author = {Pierre Baumann and Joel Kamnitzer and Peter Tingley},
title = {Affine {Mirkovi\'c-Vilonen} polytopes},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {113--205},
year = {2014},
publisher = {Springer Berlin Heidelberg},
address = {Berlin/Heidelberg},
volume = {120},
doi = {10.1007/s10240-013-0057-y},
mrnumber = {3270589},
zbl = {1332.17012},
language = {en},
url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-013-0057-y/}
}
TY - JOUR AU - Pierre Baumann AU - Joel Kamnitzer AU - Peter Tingley TI - Affine Mirković-Vilonen polytopes JO - Publications Mathématiques de l'IHÉS PY - 2014 SP - 113 EP - 205 VL - 120 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-013-0057-y/ DO - 10.1007/s10240-013-0057-y LA - en ID - PMIHES_2014__120__113_0 ER -
%0 Journal Article %A Pierre Baumann %A Joel Kamnitzer %A Peter Tingley %T Affine Mirković-Vilonen polytopes %J Publications Mathématiques de l'IHÉS %D 2014 %P 113-205 %V 120 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-013-0057-y/ %R 10.1007/s10240-013-0057-y %G en %F PMIHES_2014__120__113_0
Pierre Baumann; Joel Kamnitzer; Peter Tingley. Affine Mirković-Vilonen polytopes. Publications Mathématiques de l'IHÉS, Volume 120 (2014), pp. 113-205. doi: 10.1007/s10240-013-0057-y
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