BFV-BRST analysis of the classical and quantum q-deformations of the sl(2) algebra

BFV-BRST charge for quantum algebras is not unique. Different constructions of it in classical and quantum phase space for the universal enveloping algebra of the q-deformed sl(2) are discussed. In the quantum framework a positive definite scalar product is used to introduce a co-BFV-BRST charge to study the cohomology problem by means of the techniques of the non-deformed case. Moreover, deformation of the phase space without deforming the generators of sl(2) is considered. h(cross)-q-deformation of the phase space is shown to yield the Witten's second deformation for sl(2). To study the BFV-BRST cohomology problem when both the quantum phase space and the group are deformed, a two-parameter deformation of sl(2) is proposed, and its BFV-BRST charge is given.