Dirac operator and a twisted cyclic cocycle on the standard Podles quantum sphere

Abstract

A Dirac operator D on the standard Podleś sphere S 2 q is defined and investigated. It yields a real spectral triple such that |D| −z is of trace class for Re z > 0. Commutators with the Dirac operator give the distinguished 2dimensional covariant differential calculus on S 2 q . The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.

A residue formula for the fundamental Hochschild class on the Podleś sphere

“…[1,3,5,6,8,12,17,18,19,21,25] and the references therein. But still some basic questions remain untouched, e.g.…”

“…The pioneering papers on the noncommutative geometry of the Podleś sphere were [17], where Masuda, Nakagami and Watanabe computed HH • (A), HC • (A) and the K-theory of the C * -completion of A, and [8], where Dąbrowski and Sitarz found the spectral triple that we use here. Schmüdgen and the second author then gave a residue formula for a cyclic cocycle [21] that looks like the one from Theorem 1, only that K −2 is replaced by K 2 . However, Hadfield later computed the Hochschild and cyclic homology of A with coefficients in σ A and deduced that the cocycle from [21] is trivial as a Hochschild cocycle [9].…”

“…Schmüdgen and the second author then gave a residue formula for a cyclic cocycle [21] that looks like the one from Theorem 1, only that K −2 is replaced by K 2 . However, Hadfield later computed the Hochschild and cyclic homology of A with coefficients in σ A and deduced that the cocycle from [21] is trivial as a Hochschild cocycle [9]. Finally, the first author has recently used the cup and cap products between the Hochschild (co)homology groups H n (A, σ A) [14] to produce the surprisingly simple formula…”

“…From this fact it will be deduced in the final section that the multilinear functional defined in Theorem 1 is up to normalisation indeed cohomologous to the Hochschild cocycle (2) constructed in [14]. The remainder of the paper is divided into two sections: Section 2 contains background material taken mainly from [8,21]. The subsequent section discusses the meromorphic continuation of the zeta functions tr H ± (aK 2µ |D| −z ), a ∈ A, and how one can compute their residues by replacing certain algebra elements of A by simpler operators.…”

“…(2)). We retain all notations and conventions used in [21]. In particular, we fix a deformation parameter q ∈ (0, 1), and let A = O(S 2 q ) be the * -algebra (over C) with generators A = A * , B and B * and defining relations…”