From Quantum Universal Enveloping Algebras to Quantum Algebras

TL;DR

This paper proposes a new approach to understanding quantum groups by deriving a unique set of quantum algebra generators from Lie bialgebras, simplifying their classification and providing explicit examples.

Contribution

It introduces the analyticity-based method to define quantum algebra generators from Lie bialgebras, linking local structure to Lie bialgebra classification.

Findings

Unique quantum algebra generators derived from Lie bialgebras.

The local structure of quantum groups is represented by the analytically prolonged Lie bialgebra.

Explicit cases of su_q(2) and su_q(3) demonstrate the approach.

Abstract

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