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

TL;DR

This paper introduces the theory of braided groups and matrices, generalizing supersymmetry with braid statistics, and explores their applications in quantum groups, braided geometry, and particle physics.

Contribution

It systematically develops the algebraic framework of braided groups and matrices, extending supersymmetry and quantum group theory with new braid statistics and geometric concepts.

Findings

Braided groups generalize supersymmetry and quantum groups.

Applications include covariant tensor products, spin chains, and q-Minkowski space.

Braided geometry encompasses quantum geometry with braided derivatives.

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 $\Psi$, 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-Poincar\'e 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.