QUANTUM DUALITY CLASSICAL AND QUASI-CLASSICAL LIMITS

Quantum duality principle is applied to a study of classical limits of quantum algebras and groups. For a certain type of Hopf algebras the explicit procedure to construct both classical limits is presented. Th e canonical forms of quantized Lie-bialgebras are proved to be two-par ametric varieties with two classical limits called dual. When consider ed from the point of view of quantized symmetries, such varieties can have boundaries that are noncommutative and noncocommutative. In this case the quantum duality and dual limits still exist while instead of Lie bialgebra one has a pair of tangent vector fields.