Spin geometry on quantum groups via covariant differential calculi

Abstract

Let 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 S 2. 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 SL q (2) and its bicovariant 4 D ±-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.