Bethe algebras

Bethe代数是一类重要的交换子代数，它存在于多项式流代数的泛包络代数、Yangians代数、仿射量子群等无限维代数中。Bethe代数研究的最新进展导致了B的证明。和M.夏皮罗猜想，横截性猜想，舒伯特演算的真实性，gl（N）Gaudin模型的谱的简单性，海森堡XXX链的简单性，Bethe向量和Fuchsian无单值算子作为几何Langlands程序的一部分的对应性等等。PI希望用于获得这些结果的想法可以系统化，并进一步发展为可以应用于许多其他重要案例的程序。特别是，PI打算研究Bethe子代数在较大代数的自然表示中的图像，通过将其识别为适当代数簇上的正则函数环，然后结合表示论和代数几何的信息。PI打算建立和研究代数几何和可积模型理论之间新的密切关系。这种联系预计将在这两个学科产生各种新的成果。

The Bethe algebras are important commutative subalgebras found in many infinite-dimensional algebras such as universal enveloping algebras of polynomial current algebras, Yangians, affine quantum groups, etc. Recent progress in the study of the Bethe algebras led to the proofs of the B. and M. Shapiro conjecture, of the tranversality conjecture, of the reality of Schubert Calculus, of the simplicity of the spectrum of the gl(N) Gaudin model, of the simplicity of the Heisenberg XXX chain, of the correspondence of the Bethe vectors and Fuchsian monodromy-free operators as part of the geometric Langlands program and others. The PI expects that the ideas used to obtain these results can be systematized and further developed to a procedure which can be applied to many other important cases. In particular, the PI intends to study the images of the Bethe subalgebras in natural representations of the larger algebras by identifying it with a ring of regular functions on a suitable algebraic variety and then combining the information from the representation theory and algebraic geometry.PI intends to establish and study new close relations between algebraic geometry and theory of integrable models. Such a connection is expected to produce a variety of new results in both disciplines.