Quantum Integrable Model of an Arrangement of Hyperplanes
Quantum Integrable Model of an Arrangement of Hyperplanes
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超平面排列的量子可积模型
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
A. Varchenko
中科院分区：
文献类型：
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作者：
A. Varchenko
本文的目标是给出与简单李代数相关的高丁模型的 Bethe 代数（哈密顿量）的几何构造。更准确地说，在本文中，量子可积模型被分配给仿射超平面的加权排列。我们证明（在某些假设下）模型的哈密顿量代数与相应主函数的临界集上的函数代数同构。对于判别排列，我们表明（在某些假设下）哈密顿量代数的对称部分与相应高丁模型的贝特代数同构。预计这种对应关系总体上是成立的（没有假设）。作为构造的副产品，我们表明在高丁模型（与任意简单的李代数相关）中，对应于主函数的孤立临界点的 Bethe 向量是非零的。
The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero.