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Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra. (English) Zbl 0651.17008
For any simple complex Lie algebra finite dimensional representations of its quantum \(t\)-analogue (a deformation of its universal enveloping algebra with parameter \(t\)) are proven to be completely reducible if t is not a root of 1. For the irreducible ones there is proven a highest weight theorem and, moreover, they are shown to be deformations of the representations of the initial enveloping algebra. For \(\text{sl}(2)\) the result matches that by A. V. Odesskii [An analogue of the Sklyanin algebra, Funkts. Anal. Prilozh. 20, No. 2, 78–79 (1986; Zbl 0606.17013)].
Reviewer: D.A.Leites

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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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