Quantum Groups and Their Primitive Ideals

" By a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat­ egory of (! modules. This means of course just defining an algebra structure on Rq[G] but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an ""R-matrix"" satisfying in particular th e quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed but V. G. Dr infeld found that it could be essentially made to work for the ""Borel part"" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of t he mo st remarkable aspects of the above superficially different ap­ proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence ""canonical"", objects Rq[G] and Uq(g), though this ep ithet ma y as yet be premature. "