A Poincaré–Birkhoff–Witt theorem for generalized Lie color algebras

Abstract

A proof of the 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 An and Mp,q,ε(n,K), which is a nonstandard 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.