arXiv math/01122782001-12-26math.QA

几何晶体与量子杨-巴克斯特方程的集合论解

Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation

Pavel Etingof

备注：: 10 pages

摘要：

几何晶体的概念是由A.Berenstein和D.Kazhdan引入的，其动机是p-adic群的表示论的需要。A.Braverman、A.Berenstein和D.Kazhdan证明，某些特殊的几何晶体会产生n维环面的笛卡尔平方的有趣的双有理自同构R，它满足量子杨-巴克斯特方程和么正性条件。另一方面，Schedler、Soloviev和作者研究了量子杨-巴克斯特方程的酉集合论解。结果表明，该理论是特别好，如果解决方案满足一个额外的非退化条件。特别地，在这种情况下，可以定义所谓的简化结构群，其复杂性表征解的复杂性。本文证明了映射R是非退化的，并给出了它满足量子杨-巴克斯特方程和么正性条件的新证明。然后计算了R的约化结构群，证明了它是“loop群”PGL（n，C（t））的子群。本文还给出了Braverman和Kazhdan关于矩阵空间上两个对称群作用的交换性定理的一个新的直接证明。

英文摘要：

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.

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