Geometric and unipotent crystals
| dc.creator | Berenstein, Arkady | |
| dc.creator | Kazhdan, David | |
| dc.date | 1999-12-13 | |
| dc.date | 2006-03-29 | |
| dc.date.accessioned | 2026-07-07T06:34:32Z | |
| dc.date.available | 2026-07-07T06:34:32Z | |
| dc.description | In this paper we introduce geometric crystals and unipotent crystals which are algebro-geometric analogues of Kashiwara's crystal bases. Given a reductive group G, let I be the set of vertices of the Dynkin diagram of G and T be the maximal torus of G. The structure of a geometric G-crystal on an algebraic variety X consists of a rational morphism γ:X-->T and a compatible family e_i:G_m\times X-->X, i\in I of rational actions of the multiplicative group G_m satisfying certain braid-like relations. Such a structure induces a rational action of W on X. Quite surprisingly, many interesting rational actions of the group W come from geometric crystals. Also all the known examples of the action of W which appear in the construction of gamma-functions for the representations of ^LG in the recent work by A. Braverman and D. Kazhdan come from geometric crystals. There are many examples of positive geometric crystals on (G_m)^l, i.e., those geometric crystals for which the actions e_i and the morphism γare given by positive rational expressions. To each positive geometric crystal X we associate a Kashiwara's crystal corresponding to the Langlands dual group ^LG. An emergence of ^LG in the "crystal world" was observed earlier by G. Lusztig. Another application of geometric crystals is a construction of trivialization which is an W-equivariant isomorhism X-->γ^{-1}(e) \times T for any geometric SL_n-crystal. Unipotent crystals are geometric analogues of normal Kashiwara crystals. They form a strict monoidal category. To any unipotent crystal built on a variety X we associate a certain gometric crystal. | |
| dc.description | AMS LaTeX, 39 pages. Some typos are corrected | |
| dc.identifier | https://arxiv.org/abs/math/9912105 | |
| dc.identifier | http://arxiv.org/abs/math/9912105 | |
| dc.identifier | GAFA, Special Volume, Part I (2000), pp. 188--236 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/99553 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Representation Theory | |
| dc.title | Geometric and unipotent crystals | |
| dc.type | text |