Beyond Supersymmetry and Quantum Symmetry (an introduction to braided groups and braided matrices)

Abstract

This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi $\pm1$ statistics between Grassmannn coordinates is now replaced by a general braid statistics $Ψ$, typically given by a Yang-Baxter matrix $R$. Most of the algebraic proofs are best done by drawing knot and tangle diagrams, yet most constructions in supersymmetry appear to generalise well. Particles of braid statistics exist and can be expected to be described in this way. At the same time, we find many applications to ordinary quantum group theory: how to make quantum-group covariant (braided) tensor products and spin chains, action-angle variables for quantum groups, vector addition on $q$-Minkowski space and a semidirect product q-Poincaré group are among the main applications so far. Every quantum group can be viewed as a braided group, so the theory contains quantum group theory as well as supersymmetry. There also appears to be a rich theory of braided geometry, more general than super-geometry and including aspects of quantum geometry. Braided-derivations obey a braided-Leibniz rule and recover the usual Jackson $q$-derivative as the 1-dimensional case.