Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1993-03-26
Physics
High Energy Physics
High Energy Physics - Theory
56 pages
Scientific paper
10.1016/0393-0440(94)90014-0
We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space $\CL$ equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$ and a Yang-Baxter operator $\Psi:\CL\tens\CL\to \CL\tens\CL$ obeying some axioms. We show that such an object has an enveloping braided-bialgebra $U(\CL)$. We show that every generic $R$-matrix leads to such a braided Lie algebra with $[\ ,\ ]$ given by structure constants $c^{IJ}{}_K$ determined from $R$. In this case $U(\CL)=B(R)$ the braided matrices introduced previously. We also introduce the basic theory of these braided Lie algebras, including the natural right-regular action of a braided-Lie algebra $\CL$ by braided vector fields, the braided-Killing form and the quadratic Casimir associated to $\CL$. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations $U_q(g)$ are understood as the enveloping algebras of such underlying braided Lie algebras with $[\ ,\ ]$ on $\CL\subset U_q(g)$ given by the quantum adjoint action.
No associations
LandOfFree
Quantum and Braided Lie Algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Quantum and Braided Lie Algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quantum and Braided Lie Algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-133972