Quantum and Braided Lie Algebras

Abstract

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 $Ψ:\CL\tens\CL\to \CL\tens\CL$ 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.

56 pages

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• Quantum group gauge theory on quantum spaces
• Differential Geometry on Linear Quantum Groups
• Quantum and Braided Linear Algebra
• Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map
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• The van Est spectral sequence for Hopf algebras
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• q-Deformed Superalgebras
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• Equivariant quantizations of the positive nilradical and covariant differential calculi
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