Regularity and dimension spectrum of the equivariant spectral triple for the odd-dimensional quantum spheres

Abstract

Abstract. The odd-dimensional quantum sphere S 2`C1 q is a homogeneous space for the quantum group SU q .`C 1/. A generic equivariant spectral triple for S 2`C1 q on its L 2 -space was constructed by Chakraborty and Pal in [4]. We prove regularity for that spectral triple here. We also compute its dimension spectrum and show that it is simple. We give a detailed construction of its smooth function algebra and some related algebras that help proving regularity and in the computation of the dimension spectrum. Following the idea of Connes for SU q .2/, we first study another spectral triple for S 2`C1 q equivariant under torus group action and constructed by Chakraborty and Pal in [3]. We then derive the results for the SU q .`C 1/-equivariant triple in the case q D 0 from those for the torus equivariant triple. For the case q ¤ 0, we deduce regularity and dimension spectrum from the case q D 0. (2010). 58B34, 46L87, 19K33. Mathematics Subject Classification

Asymptotic and Exact Expansions of Heat Traces

“…For brevity we shall call such operators -the operators of polynomial spectrum. They appear naturally in the context of Dirac and Laplace operators on spheres [6,69] and their isospectral deformations [17,18,[24][25][26]62]. Moreover, the results presented in this section apply almost directly when the spectra of the relevant operators can be written as…”

Quantum double suspension and spectral triples

“…In particular we obtain the regularity and dimension spectrum of the odd-dimensional quantum spheres. This gives a conceptual explanation of the results obtained in [14].…”

“…Hence by Theorem 3.2 this is regular with finite dimension spectrum. This gives a conceptual proof of Proposition 3.9 in [14].…”

“…We refer to [12] or Lemma 3.2 of [14] for the proof. Thus the odd-dimensional quantum spheres can be obtained from the circle T by applying the quantum double suspension recursively.…”

“…The first simple illustration was given by Connes in [7]. This was extended to odd-dimensional quantum spheres by Pal and Sundar in [14]. But to have a good grasp of the formula it is essential to have a systematic family of examples where one can verify the hypothesis of regularity and discreteness of the dimension spectrum.…”