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 the 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. Drinfeld 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 the most 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 epithet may as yet be premature."
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