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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:
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