Alberto De Sole
Lie Algebras, representation theory
Vertex algebras, Poisson vertex algebras, integrable Hamiltonian systems, W-algebras
In the theory of integrable systems a key role is covered by Hamiltonian, or bi-Hamiltonian, equations.
Indeed, the presence of two compatible Hamiltonian structures for the same evolution equation is intrinsically related to the existence of infinitely many integrals of motion in involution.
A rigorous theory of integrable Hamiltonian partial differential equations has been developed, based on the notion of a Poisson vertex algebra. In particular, following the main idea of [BDSK09], one ca apply the Lenard-Magri scheme to find new integrable bi-Hamiltonian equations, and possibly to find a complete classification in the scalar case.
W-algebras play a special role both in the representation theory of Lie algebras and vertex algebras, and in the theory of integrable systems. An important problem is to find explicit generators and relations for the quantum affine W-algebra W(g,f) attached to a simple finite dimensional Lie algebra g and any its nilpotent element f, and to study the corresponding (quantum) integrable Hamiltonian systems.
[DS85] Drinfeld V.G., Sokolov V.V., Lie algebras and equations of KdV type, Soviet J. Math. 30 (1985), 1975-2036
[Dick03] Dickey L. A., Soliton equations and Hamiltonian systems, Advanced series in mathematical physics, World scientific, Vol. 26, 2nd Ed., 2003
[BDSK06] Barakat A., De Sole A., Kac V. G., Poisson vertex algebras in the theory of Hamiltonian equations, Japan. J. Math. 4 (2009), n.2, 141-252
[DSKV13] De Sole A., Kac V. G., Valeri D., Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras, Comm. Math. Phys. 323 (2013), n.2, 663-711
[DSKV] De Sole A., Kac V. G., Valeri D., Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras, Comm. Math. Phys. 323 (2013), n.2, 663-711