2001/12/26 by Pavel Etingof, Etingof, Pavel
Computer Science · Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Quantum Algebra (math.QA) #Topological and Geometric Data Analysis #advanced mathematical theories #math.QA
paper · pdf · doi:10.48550/arxiv.math/0112278
10 pages
arxiv created 2001/12/26 · openalex publication_date 2001/12/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of a geometric crystal was introduced by A.Berenstein and D.Kazhdan, motivated by the needs of representation theory of p-adic groups. It was shown by A.Braverman, A.Berenstein, and D.Kazhdan that some particular geometric crystals give rise to an interesting birational automorphism R of the Cartesian square of an n-dimensional torus, which satisfies the quantum Yang-Baxter equation and the unitarity condition. On the other hand, unitary set-theoretical solutions of the quantum Yang-Baxter equation were studied by Schedler, Soloviev, and the author. It was shown that the theory is especially nice if the solution satisfies an additional nondegeneracy condition. In particular, in this situation one can define the so called reduced structure group, whose complexity characterizes the complexity of the solution. In this note we show that the map R is nondegenerate, and give a new proof that it satisfies the quantum Yang-Baxter equation and the unitarity condition. Then we calculate the reduced structure group of R, and show that it is a subgroup of the "loop group" PGL(n,C(t)). We also give a new, direct proof of a Theorem of Braverman and Kazhdan on the commutativity of two symmetric group actions on the space of matrices.