zbMATH — the first resource for mathematics

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.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
Full Text: DOI Numdam EuDML
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.