Title:
Hidden quantum group structure on solution spaces of certain PDE Systems
Abstract:
I describe a systematic method for solving PDEs of certain type,
which arise in conformal field theory, and in the theory of Schramm-Loewner
evolutions, with boundary conditions given by specified asymptotic
behavior of the solutions. Our method is a correspondence associating
vectors in a tensor product representation of a quantum group to Coulomb gas
type integral functions, in which properties of the functions are encoded
in natural, representation theoretical properties of the vectors.
In particular, this hidden quantum group structure on the solution space
of such PDEs enables explicit calculation of the asymptotics and monodromy
properties of the solutions. This also leads us to a generalization of
the Temperley-Lieb algebra, defined in terms of a diagrammatic
representation. This generalized diagram algebra is the commutant
algebra of the quantum group, generalizing the quantum Schur-Weyl duality.