Equivariant Spectral Triples on the Quantum SU(2) Group

Abstract

We characterize all equivariant odd spectral triples for the quantum SU (2) group acting on its L 2 -space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SU q (2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU (2), and we prove that for p < 4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dimension p acting on the L 2 -space.

The Geometry of Noncommutative Spacetimes

“…However, the landscape of Connes noncommutative geometry is much broader. It includes various classical 'pathological' spaces, such as fractals [42], isospectral deformations [43] (in particular the celebrated noncommutative torus [7,44]) or quantum groups [45] and their homogeneous spaces [46].…”

Index Theory and Adiabatic Limit in QFT

“…In fact if we want to describe the Galilean spacetime spectrally and explain in addition its connection to non-relativistic quantum fields, it is the central Bargmann extension 39 G of the inhomogeneous Galilean group which is more natural here as a symmetry group. Indeed the appropriate Dirac operator 40 we should use here is the non relativistic Dirac operator −i∂ t ⊗ A − i∂ i ⊗ B i + 1 ⊗ C found by Lévy-Leblond [27], where A, B i , C are elements of a Clifford algebra over the (five dimensional) extension 41 of tangent space with a positive definite and singular quadratic form in it. Indeed, in this 37 Although action of the extension on the Galilean spacetime degenerates to the ordinary action of the inhomogeneous Galilean group, using of the Bargmann extension is essential if one intents to describe spectrally the Galilean spacetime manifold, see below for some further comments.…”

“…40 Although for general Dirac-type operator on M with a pseudo-riemannian metric, or even with a more degenerate "metric" structure, there does not exist any natural Hilbert space acted on by D, such that D is (essentially) selfadjoint, there nevertheless does exist natural Krein-type space with respect to which the operator D is selfadjoint in the generalized Krein sense, compare e.g. [26].…”

“…Quite recent works show that the two formalisms may be reconciled. Let us cite the breakthrough papers only [40][41][42][43].…”

Equivariant spectral triples and Poincaré duality for 𝑆𝑈_{𝑞}(2)

“…If one replaces condition 3 above with a slightly weaker condition, then one says that the spectral triples (H, π, D) and (H, π , D) are rational Poincaré duals (see [12] for this notion). In an earlier paper ( [3]), the authors constructed an equivariant spectral triple for the quantum SU (2) group that was later analysed further by Connes in [8]. It is natural to ask whether the triple gives rise to a fundamental class for SU q (2).…”