Killing Form on Quasitriangular Hopf Algebras and Quantum Lie Algebras

Abstract

The basics of quasitriangular Hopf algebras and quantum Lie algebras are briefly reviewed, and it is shown that their properties allow the introduction of a Killing form. For quantum Lie algebras, this leads to the definitions of a Killing metric and quadratic casimir. The specific case of $\uea{su}{N}$ is examined in detail, where it is shown that many of the classical results are reproduced, and explicit calculations to illustrate the conclusions are presented for $\uea{su}{2}$.