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Quantum deformations of certain simple modules over enveloping algebras. (English) Zbl 0651.17007
Let $$A$$ be a symmetrizable Cartan matrix. Drinfeld and Jimbo have associated to A and to any field $$F$$ a Hopf algebra $$\hat U$$ (called sometimes ”quantum group”) depending on a parameter $$q$$ in $$F$$. The universal enveloping algebra of the Kac-Moody Lie algebra over a subfield $$F_ 0$$ of F corresponding to $$A$$ can be obtained as a limit of $$\hat U$$ as $$q$$ tends to 1. The main result of the paper is that (when $$\text{char}(F) = 0$$ and $$\det(A) = 0$$) any simple integrable highest weight module V of U admits a ”quantum deformation”, i.e. there is a simple $$\hat U$$-module $$\hat V$$ such that $$\hat V$$ tends to $$V$$ as $$q$$ tends to 1. Under an additional condition on $$A$$, an action of the braid group on $$\hat U$$ is described.
Reviewer: L.Vaserstein

##### MSC:
 17B35 Universal enveloping (super)algebras 17B65 Infinite-dimensional Lie (super)algebras 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
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