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 selfdual (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 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.

Noncommutative Kähler structures on quantum homogeneous spaces

“…The classical extension uses anti-commutativity of forms in a fundamental way, while the multiplicative relations for a differential calculus over a noncommutative algebra will in general be much more badly behaved. In certain cases, such as bicovariant calculi [40], or braided complex structures [21], one can formulate a braided generalisation of the classical construction [12]. However, in practice the metrics produced are not ideal [21].…”

Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces

“…Let us mention here that other approaches to noncommutative field theory have been developped. The Connes spectral action [7] (also see [32] and references therein for an updated review) relies on a definition of the differential calculus based on the notion of spectral triples, while a quantum group gauge theory on quantum spaces (see [6,4,24]) is based on the notion of covariant calculi (see [47,48]). We shall not discuss further these approaches in the present article, since the algebras we consider come as subalgebras of the well known Moyal four dimensional one.…”

A Hodge–de Rham Dirac operator on the quantum SU(2)

“…This approach is evolved in [1,2,14] for quantum groups equipped with a Woronowicz bicovariant exterior calculus, thus allowing for the definition of a Dirac operator. The papers [12,13,16] develop a consistent formulation of the notions of Clifford algebras for quantum groups equipped with left covariant Woronowicz calculi: the corresponding exterior algebras are left modules for the Clifford algebra (generalising (1.4)), spinors are introduced algebraically in terms of irreducibility subspaces of such an action. A consistent notion of metric tensors and Levi Civita connection acting upon exterior forms is introduced, a quantum analogue of the Hodge -de Rham Dirac operator generalising (1.9) is defined and studied.…”