Algebraic structures arising in physics

这项提议主要围绕以下五个相互关联的主题。第一个主题是关于有理W-代数和模不变表示。量子哈密顿约化已经发展成为解决许多来自物理学和表象理论的公开问题的有力工具。下一个目标是有理W-代数的分类，它应该在共形场理论中发挥重要作用。其基本思想是利用仿射李代数的模不变表示。第二个主题是关于四个基本的代数结构。这是一个一般的框架，它将Poisson代数、结合代数、Poisson顶点代数和顶点代数统一在一幅图中，分别强调经典力学、量子力学、经典场论和最简单的量子场论。第三个主题是关于自由生成的单点代数和Poisson顶点代数的分类。讨论了关于自由有限生成顶点代数和Poisson顶点代数分类的一个猜想。“无限”问题被归结为一个有限但“非线性”的问题。对后者的计算机搜索证实了这一猜想。第四个主题是关于Poisson顶点代数和无限维可积哈密顿系统。Poisson顶点代数是哈密顿偏微分方程论最合适的语言。这一观点进一步发展了发展方程的可积性理论。最后一个主题是关于李共形代数、上同调和变分。目的是研究经典变分复形与李共形代数的上同调理论之间的关系。这种联系应该会导致这两个理论的进一步发展。预计这一建议将对数学的几个分支产生重大的统一影响，可能还会对理论物理学产生影响。例如，经典和量子W-代数理论的发展应该导致经典和量子可积系统理论的进一步发展。四个基本代数结构的理论一方面可以加深对表示论、拓扑学和孤立子理论与物理理论四个基本框架之间的联系的理解。Poisson顶点代数的理论应该导致可积系理论的进一步发展，如KdV族和非线性薛定谔族。李共形代数的上同调理论应该会导致对变分的更好的理解，这一理论可以追溯到欧拉和拉格朗日的工作。

This proposal is concentrated around the following five interrelatedtopics. The first topic is on rational W-algebras and modular invariant representations. Quantum Hamiltonian reduction has been developed as a powerful tool for solving many open problems coming from physics and representation theory. The next goal is classification of rational W-algebras, which should play an important role in Conformal field theory. The basic idea is to utilize modular invariant representations of affine Lie algebras. The second topic is on four fundamental algebra structures. This is a general framework, which unites in one picture Poisson algebras, associative algebras, Poisson vertex algebras and vertex algebras, that underline classical mechanics, quantum mechanics, classical field theory, and the simplest quantum field theory, respectively. The third topic is on classification of freely generated simple vertex and Poisson vertex algebras. A conjecture on classification of freely and finitely generated vertex algebras and Poisson vertex algebras is discussed. The "infinite" problem is reduced to a finite, but "non-linear'" problem. A computer search for the latter has confirmed the conjecture.The fourth topic is on Poisson vertex algebras and infinite-dimensional integrable Hamiltonian systems. Poisson vertex algebras is the most adequate language for the theory of Hamiltonian PDE. This point of view leads to further development of the integrability theory of evolutionary equations. The last topic is on Lie conformal algebra cohomology and calculus of variations. The goal is to study the relation of the classical variational complex to the cohomology theory of Lie conformal algebras. This connection should lead to further development of both theories.It is expected that this proposal will have a significant unifying impact upon several branches of mathematics and, possibly, upon theoretical physics. For example, the development of the theory of classical and quantum W-algebras should lead to further progress in the theory of classical and quantum integrable systems. The theory of four fundamental algebraic structures should lead to deeper understanding of connections between representation theory, topology and soliton theory on the one hand and the four fundamental frameworks of physics theories on the other hand. The theory of Poisson vertex algebras should lead to further development of the theory of integrable systems, such as the KdV hierarchy and the non-linear Schrödinger hierarchy. The cohomology theory of Lie conformal algebras should lead to a better understanding of the variational calculus, a theory that goes back to the works of Euler and Lagrange.