From quantum universal enveloping algebras to quantum algebras

Abstract

The ‘local’ structure of a quantum group Gq is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra Uq(g), which is a Hopf algebra deformation of the universal enveloping algebra of an n-dimensional Lie algebra g = Lie(G). However, we show how, by starting from the generators of the underlying Lie bialgebra (g, δ), the analyticity in the deformation parameter(s) allows us to determine in a unique way a set of n ‘almost primitive’ basic objects in Uq(g), which could be properly called the ‘quantum algebra generators’. So, the analytical prolongation (gq, Δ) of the Lie bialgebra (g, δ) is proposed as the appropriate local structure of Gq. Besides, as in this way (g, δ) and Uq(g) are shown to be in one-to-one correspondence, the classification of quantum groups is reduced to the classification of Lie bialgebras. The suq(2) and suq(3) cases are explicitly elaborated.