Spin geometry on quantum groups via covariant differential calculi

AbstractLet A be a cosemisimple Hopf ∗-algebra with antipode S and let Γ be a left-covariant first-order differential ∗-calculus over A such that Γ is self-dual (see Section 2) and invariant under the Hopf algebra automorphism S2. A quantum Clifford algebra Cl(Γ,σ,g) is introduced which acts on Woronowicz’ external algebra Γ∧. A minimal left ideal of Cl(Γ,σ,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 Γ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SLq(2) and its bicovariant 4D±-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