Skew-Primitive Elements of Quantum Groups and Braided Lie Algebras

Abstract

In the study of Lie groups, of algebraic groups or of formal groups, the concept of Lie algebras plays a central role. These Lie algebras consist of the primitive elements. It is difficult to introduce a similar concept for quantum groups. Many important quantum groups have braided Hopf algebras as building blocks. Most primitive elements live in these braided Hopf algebras. This chapter provides a survey of and a motivation for this concept together with some interesting examples. The category of Yetter-Drinfel'd modules is in a sense the most general category of modules carrying a natural braiding on the tensor power of each module (instead of a symmetric structure). Quantum groups arise from deformations of universal enveloping algebras of Lie algebras. The chapter also provides two examples which show how to construct large families of Lie algebras from Yetter-Drinfel’d algebras and from Yetter-Drinfel’d modules with a bilinear form in a similar way as one does for classical Lie algebras.