Generalized Lie Algebras and Cocycle Twists

Abstract

We use the results of Etingof and Gelaki on the classification of (co)triangular Hopf algebras to extend Scheunert's “discoloration” technique to Lie algebras in the category of (co)modules. As an application, we prove a PBW-type theorem for such Lie algebras. We also discuss the relationship between Lie algebras in the category of (co)modules and symmetric braided Lie algebras introduced by Gurevich. Finally, we construct examples of symmetric braided Lie algebras that are essentially different from Lie coloralgebras.

Key Words:

• Braiding
• Cocycle
• Coloralgebra
• Hopf algebra
• Lie algebra
• Superalgebra

2000 Mathematics Subject Classification:

• 16W30
• 16W55
• 17B37
• 17B75

ACKNOWLEDGMENT

The author acknowledges the support of NSERC Postdoctoral Fellowship in 2003–2005. The author would like to thank Prof. Susan Montgomery for helpful discussions and the University of Southern California for hospitality.

Notes

Communicated by V. A. Artamonov.