Integrals of motion and quantum groups

We give a homological construction of integrals of motion in classical and quantum Toda field theories. This allows to find the spaces of integrals of motion by computing the cohomologies of certain complexes, closely connected with the Bernstein-Gelfand-Gelfand resolutions of the associated Lie algebras and their quantum deformations. This way we reprove the standard results on the classical integrals of motion and then prove that all of them can be quantized. For the Toda field theories associated to finite-dimensional Lie algebras, the algebra of integrals of motions coincides with the corresponding W-algebra. For affine Toda field theories this algebra is a commutative subalgebra of a W-algebra, and it consists of the hamiltonians of a generalized KdV hierarchy or their quantum deformations.

lectures: Montecatini Terme 1993/06/14
field theory: Toda
field theory: affine
dimension: 2
integrability
conservation law
quantum group
Hamiltonian formalism
operator: algebra
algebra: Lie

References(0)

Figures(0)

Loading ...