A Poincaré-Birkhoff-Witt theorem for generalized color Lie algebras
arXiv:q-alg/9706016 · doi:10.1063/1.532471
Abstract
A proof of Poincaré-Birkhoff-Witt theorem is given for a class of generalized Lie algebras closely related to the Gurevich S-Lie algebras. As concrete examples, we construct the positive (negative) parts of the quantized universal enveloping algebras of type A_n and M_{p,q,e}(n,K), which is a non-standard quantum deformation of GL(n). In particular, we get, for both algebras, a unified proof of the Poincaré-Birkhoff-Witt theorem and we show that they are genuine universal enveloping algebras of certain generalized Lie algebras.
19 pages, AMSLaTeX, 9 figures. In this version there are some notation changes. Remark II.1 corrected. A new definition (Def. VII.2) added, in order to include the supercase. New acknowledgments added. Submitted to the J. Math. Phys