Quantum and Braided Lie Algebras

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.Comment: 56 page

Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map

Braided differential operators \del^i are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang-Baxter matrix $R$. The quantum eigenfunctions \exp_R(\vecx|\vecv) of the \del^i (braided-plane waves) are introduced in the free case where the position components $x_i$ are totally non-commuting. We prove a braided $R$-binomial theorem and a braided-Taylors theorem \exp_R(\veca|\del)f(\vecx)=f(\veca+\vecx). These various results precisely generalise to a generic $R$-matrix (and hence to $n$-dimensions) the well-known properties of the usual 1-dimensional $q$-differential and $q$-exponential. As a related application, we show that the q-Heisenberg algebra $px-qxp=1$ is a braided semidirect product \C[x]\cocross \C[p] of the braided line acting on itself (a braided Weyl algebra). Similarly for its generalization to an arbitrary $R$-matrix.Comment: 19 page

Hopf (Bi-)Modules and Crossed Modules in Braided Monoidal Categories

Hopf (bi-)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. As a consequence a generalization of the Radford-Majid criterion for a braided Hopf algebra to be a cross product is obtained. The results of this paper turn out to be fundamental for the construction of (bicovariant) differential calculi on braided Hopf algebras.Comment: uuencoded compressed postscript file, 20 page