Quantum Groups and Their Primitive Ideals PDF

by Anthony Joseph

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathemati series

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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.