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

Details A Poincaré-Birkhoff-Witt theorem for generalized color Lie algebras A Poincaré-Birkhoff-Witt theorem for generalized color Lie algebras

1997-06-15

arxiv.org/abs/q-alg/9706016v4

J. Math. Phys. 39 (1998) 3828-3843

Mathematics

Quantum Algebra

19 pages, AMSLaTeX, 9 figures. In this version there are some notation changes. Remark II.1 corrected. A new definition (Def.

Scientific paper

10.1063/1.532471

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.

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