Infinite-dimensional Lie algebras and their applications

李理论以挪威数学家索夫斯·李命名，是现代数学中一个充满活力的领域，其应用范围从弦理论到卫星控制和量子计算。提出了仿射代数变体上向量场李代数的表示理论。这个理论结合了交换代数和非交换代数的特点。基于我们在环面上向量场表示理论方面的最新进展，我们想要证明一个在任意光滑代数变集下的分类定理。这个建议的一部分是更实用的。我们提出解决几个酉群上的最优控制问题。解决这些问题将对量子控制和量子计算产生影响。

Lie theory, named after a Norwegian mathematician Sophus Lie, is a vibrant area of modern mathematics with applications ranging from string theory to satellite control and quantum computing. We propose to develop representation theory of Lie algebras of vector fields on affine algebraic varieties. This theory will combine the features of commutative and non-commutative algebra. Building on the recent advances we made in representation theory of vector fields on a torus, we would like to prove a classification theorem in the setting of an arbitrary smooth algebraic variety. A part of this proposal is of a more applied nature. We propose to solve several optimal control problems on unitary groups. Solving these problems will have ramifications for quantum control and quantum computing.