Abstract. As a natural generalization of ordinary Lie algebras we introduce the concept of quantum Lie algebras . We define these in terms of certain adjoint submodules of quantized enveloping algebras endowed with a quantum Lie bracket given by the quantum adjoint action. The structure constants of these algebras depend on the quantum deformation parameter q and they go over into the usual Lie algebras when q = 1. The notions of q-conjugation and q-linearity are introduced. q-linear analogues of the classical antipode and Cartan involution are defined and a generalized Killing form, q-linear in the first entry and linear in the second, is obtained. These structures allow the derivation of symmetries between the structure constants of quantum Lie algebras. The explicitly worked out examples of and illustrate the results.

Print publication: Issue 8 (21 April 1996)
Received 7 November 1995, in final form 15 January 1996