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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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References:
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