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


[1] Borel, A.: On the complete reducibility of linear complex semi-simple Lie algebras. Private communication
[2] Drinfeld, V. G.: Hopf algebras and the quantum Yang-Baxter equation. Sov. Math. Dokl.32, 254-258 (1985)
[3] Drinfeld, V. G.: Quantum groups. Proc. I.C.M. Berkeley, 1986
[4] Humphereys, J. E.: Introduction to Lie algebras and Representation Theory. Graduate Texts in Mathematics Vol.9. Berlin, Heidelberg, New York: Springer
[5] Jimbo, M.: Aq-difference analog of 593-1 and the Yang-Baxter equation. Lett. Math. Phys.10, 63-69 (1985) · Zbl 0587.17004
[6] Jimbo, M.: Aq-analog ofU (gl(N+1)), Hecke algebras and the Yang-Baxter equation. Lett. Math. Phys.11, 247-252 (1986) · Zbl 0602.17005
[7] Rosso, M.: Représentation irréductibles de dimension finie duq-analogue de l’algèbre enveloppante d’une algebre de Lie simple. C.R.A.S. Paris. t.305. Série I. 587-590 (1987) · Zbl 0624.17005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.