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

Conceptual Issues for Noncommutative Gravity on Algebras and Finite Sets

We discuss some of the issues to be addressed in arriving at a definitive noncommutative Riemannian geometry that generalises conventional geometry both to the quantum domain and to the discrete domain. This also provides an introduction to our 1997 formulation based on quantum group frame bundles. We outline now the local formulae with general differential calculus both on the base `quantum manifold' and on the structure group Gauge transforms with nonuniversal calculi, Dirac operator, Levi-Civita condition, Ricci tensor and other topics are also covered. As an application we outline an intrinsic or relative theory of quantum measurement and propose it as a possible framework to explore the link between gravity in quantum systems and entropy.Comment: 17 pages, to appear Proc. Euroconference on Noncommutative Geometry and Hopf Algebras in Field Theory and Particle Physics, Torino, 1999 -- this intro for theoretical physicists (mathematicians, see long paper

Duality Principle and Braided Geometry

We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck scale physics. More recent work includes corresponding results at the semiclassical level of Poisson-Lie groups and at the level of braided groups and braided geometry.Comment: 24 page

Moduli of quantum Riemannian geometries on <= 4 points

We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously. The full moduli space is found for $\le 3$ points, and a restricted moduli space for 4 points. The topological part of the moduli space is found for $\le 9$ points based on the known atlas of regular graphs. We also discuss aspects of the quantum theory defined by functional integration.Comment: 34 pages ams-latex, 4 figure