Bethe algebras

贝特代数是许多无限维代数中重要的交换子代数，如多项式电流代数的普适包络代数、扬子代数、仿射量子群等。贝特代数研究的最新进展导致了B.和M.夏皮罗猜想的证明，横向猜想的证明，舒伯特微积分的实在性的证明，gl(N) Gaudin模型谱的简单性的证明，海森堡XXX链的简单性的证明，贝特向量和作为几何朗兰兹程序一部分的Fuchsian无单算子的对应性的证明等等。PI期望用于获得这些结果的想法可以系统化，并进一步发展为可应用于许多其他重要案例的程序。特别是，PI打算通过在合适的代数变体上用正则函数环识别贝特子代数，然后将表示理论和代数几何的信息结合起来，研究贝特子代数在大代数的自然表示中的象。PI旨在建立和研究代数几何与可积模型理论之间的新的密切关系。这种联系有望在这两个学科中产生各种新的结果。

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.