Infinite-dimensional representations of quantum algebras

Abstract

This chapter illustrates the characteristic properties and special features of infinite-dimensional representations of quantum algebras. Quantum groups appear as mathematical objects describing symmetries in non-commutative geometry. They are “motion groups” of such objects of non-commutative geometry as the quantum plane, quantum spheres, and quantum hyperboloids. The theory of the q-analogue of the quantum harmonic oscillator is developed. The quantum Heisenberg group is also constructed. Quantum groups and algebras are applied to the investigation of basic hypergeometric functions and q-orthogonal polynomials in the same way as Lie groups and discrete groups are used for investigation of the usual special functions and orthogonal polynomials. Some new addition theorems and product formulas for q-orthogonal polynomials are proved. The infinite-dimensional representation of the quantum algebra Uq (su l.l ), which is one of simplest noncompact quantum algebras, is described. The representations of Uq (su l.1 ) show the main features of infinite dimensional irreducible representations of quantum algebras with respect to irreducible representations of simple Lie algebras. The infinite dimensional representations of some other quantum algebras are also discussed. Quantum algebras and quantum groups are equipped with the structure of Hopf algebras.