A Poincar\'e-Birkhoff-Witt theorem for generalized color Lie algebras

Abstract

A proof of Poincar\'e-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 Poincar\'e-Birkhoff-Witt theorem and we show that they are genuine universal enveloping algebras of certain generalized Lie algebras.