Spin Geometry on Quantum Groups via Covariant Differential Calculi

Abstract

Let A be a cosemisimple Hopf *-algebra with antipode S and let $\Gamma$ be a left-covariant first order differential *-calculus over A such that $\Gamma$ is self-dual and invariant under the Hopf algebra automorphism S^2. A quantum Clifford algebra $\Cl(\Gamma,\sigma,g)$ is introduced which acts on Woronowicz' external algebra $\Gamma^\wedge$. A minimal left ideal of $\Cl(\Gamma,\sigma,g)$ which is an A-bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on $\Gamma$ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SL_q(2) and its bicovariant $4D_\pm$-calculi these concepts are studied in detail. A generalization of Bochner's theorem is given. All invariant differential operators over a given spinor module are determined. The eigenvalues of the Dirac operator are computed. Keywords: quantum groups, covariant differential calculus, spin geometry