A POINCARE-BIRKHOFF-WITT THEOREM FOR GENERALIZED LIE COLOR ALGEBRAS

A proof of the Poincare-Birkhoff-Witt theorem is given for a class of generalized Lie algebras closely related to the Gurevich S-Lie algebra s. As concrete examples, we construct the positive (negative) parts of the quantized universal enveloping algebras of type A(n) and M-p,M-q, M-epsilon(n,K), which is a nonstandard quantum deformation of GL(n). I n particular, we get, for both algebras, a unified proof of the Poinca re-Birkhoff-Witt theorem, and we show that they are genuine universal enveloping algebras of certain generalized Lie algebras. (C) 1998 Amer ican Institute of Physics. [S0022-2488(98)00406-X].