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


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


Zbl 0606.17013
Full Text: DOI


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