A residue formula for the fundamental Hochschild class of the Podles sphere

The fundamental Hochschild cohomology class of the standard Podles quantum sphere is expressed in terms of the spectral triple of Dabrowski and Sitarz by means of a residue formula.

💡 Research Summary

The paper addresses a concrete realization of the fundamental Hochschild cohomology class of the standard Podleś quantum sphere by means of the spectral triple introduced by Dabrowski and Sitarz. The Podleś sphere, denoted (\mathcal{A}(S^2_q)), is a well‑studied non‑commutative analogue of the 2‑sphere obtained from the quantum group (SU_q(2)). Its algebra is generated by elements (A, B, B^) subject to the (q)-deformed relations (AB = q^2 BA), (AB^ = q^{-2} B^*A), and a quadratic sphere relation. The Hochschild cohomology of this algebra contains a distinguished 2‑cocycle that plays the role of a volume form; this is the “fundamental Hochschild class”.

Dabrowski and Sitarz constructed a 2‑dimensional spectral triple ((\mathcal{A},\mathcal{H},D)) for (\mathcal{A}(S^2_q)). The Hilbert space (\mathcal{H}) is the direct sum of two copies of the spin‑(1/2) representation of (SU_q(2)); the Dirac operator (D) acts as a (q)-deformed differential operator with spectrum ({\pm(2k+1)\mid k\in\mathbb{N}}). The triple satisfies the usual regularity and finiteness conditions, and its “analytic dimension” is two, as shown by the meromorphic continuation of (\operatorname{Tr}(|D|^{-z})).

The core of the article is the construction of a Hochschild 2‑cocycle (\varphi) from the spectral data using the Connes‑Moscovici residue trace. For (a_0,a_1,a_2\in\mathcal{A}) the authors define
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