## 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

Zbl 0606.17013
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### References:

 [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
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