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Analogues de la forme de Killing et du théorème d’Harish-Chandra pour les groupes quantiques. (Analogues of the Killing form and of the Harish- Chandra theorem for quantum groups). (French) Zbl 0721.17012
Quantum groups are understood in the sense of Drinfel’d and Jimbo, i.e., as deformations $$U_ h{\mathfrak g}$$ of Lie algebras $${\mathfrak g}$$ over the field $${\mathbb{C}}[[h]]$$ of formal power series. The purpose of the paper is to develop suitable analogs of the concept of the Killing form and the Harish-Chandra theorem in the case where $${\mathfrak g}$$ is semisimple. With that purpose, the author concentrates on a certain $${\mathbb{C}}$$-subalgebra of $$U_ h{\mathfrak g}$$, named $$U_ t{\mathfrak g}$$, where t is a comlex parameter. An analog of the Killing form is introduced and carefully studied in most part of the paper, as a bilinear form on $$U_ t{\mathfrak g}$$ invariant under the adjoint representation. Its nondegeneracy and links with representations are treated. In the rest of the paper, the author states a q-analog of the Harish-Chandra theorem; the complete reducibility of finite-dimensional representations of $$U_ t{\mathfrak g}$$; some properties of characters of finite-dimensional irreducible $$U_ t{\mathfrak g}$$-modules.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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##### References:
 [1] A. BOREL , Communication privée. [2] V. G. DRINFLED , Quantum Groups (Proc. of I.C.M., Berkeley, 1986 ). [3] J. E. HUMPHREYS , Finite and Infinite Dimensional Modules for Semisimple Lie Algebras , dans Lie Theories and their Applications, Queen’s Papers in Pure Appl. Math., n^\circ 48. MR 80i:17009 | Zbl 0392.17006 · Zbl 0392.17006 [4] M. JIMBO , A q-Difference Analog of Ug and the Yank, Baxter Equation (Lett. Math. Phys., Vol. 10, 1985 , p. 63-69). MR 86k:17008 | Zbl 0587.17004 · Zbl 0587.17004 · doi:10.1007/BF00704588 [5] M. JIMBO , A q-Analog of U(gl(N + 1)), Hecke Algebras and the Yang-Baxter Equation (Lett. Math. Phys., vol. 11, 1986 , p. 247-252). MR 87k:17011 | Zbl 0602.17005 · Zbl 0602.17005 · doi:10.1007/BF00400222 [6] G. LUSZTIG , Quantum Deformations of Certain Simple Modules over Enveloping Algebras (Adv. Maths., vol. 70, 1988 , p. 237-249). MR 89k:17029 | Zbl 0651.17007 · Zbl 0651.17007 · doi:10.1016/0001-8708(88)90056-4 [7] M. ROSSO , Finite Dimensional Representations of the Quantum Analog of the Enveloping Algebra of a Complex Simple Lie Algebra (Comm. Math. Phys., vol. 117, 1988 , p. 581-593). Article | MR 90c:17019 | Zbl 0651.17008 · Zbl 0651.17008 · doi:10.1007/BF01218386 · minidml.mathdoc.fr [8] M. ROSSO , An Analogue of P.B.W. Theorem and the Universal R-matrix for Uhsl(N + 1) (Comm. Math. Phys., vol. 124, 1989 , p. 307-318). Article | MR 90h:17019 | Zbl 0694.17006 · Zbl 0694.17006 · doi:10.1007/BF01219200 · minidml.mathdoc.fr
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