Algebraic structures arising in physics

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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.