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

Abstract

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

Heat Trace and Spectral Action on the Standard Podleś Sphere

“…where the P 's are in the polynomial algebra D(A) generated by A, JAJ −1 , D and |D| and moreover, the ≃ means that T = N n=0 P n |D| d−n mod (OP −N ) for each N ∈ N. However A q OP 0 : it is known [35,41] that the U q (su(2))-equivariant spectral triple on standard Podleś sphere does not satisfy the regularity assumption. In fact it does not even satisfy the weak regularity condition.…”

“…Due to the convergence of series (36) and local uniform convergence of (37), the series i∈N T a L i (s) are locally uniformly convergent in each of the three cases (L = A, B, C). Hence, to show the local uniform convergence of the series (33)(34)(35), it is sufficient to check that for any s 0 ∈ C \ Sd, there exist a neighborhood U ∋ s 0 , U ∩ Sd = ∅ and constants…”

“…First pathology comes from the fact that the highest real number in the dimension spectrum is 0 hence, S 2 q is 0-summable, see (15). This drawback, already known in the literature [35,41] can be cured by the introduction of an auxiliary twist [35]. A second unexpected feature is the existence in the dimension spectrum of poles of second order.…”

Nonunital spectral triples and metric completeness in unbounded KK-theory

“…The most recent refinement in [13] was the inclusion of a class of unbounded projections into the theory, required to deal with examples arising from the Hopf fibration. Unbounded projections also appear in the construction of products for Cuntz-Krieger algebras [24], the natural Kasparov module for SU q (2) [34,46] and the differential approach to the stabilisation theorem [29]. In the previous section of the present paper, the notion of complete projective module enlarges the class of unbounded projections we can work with.…”

Quantum Dimension and Quantum Projective Spaces

“…For the case of the Podleś sphere, it is shown in [20] that such a twisted cocycle is indeed non-trivial, when D N is taken to be the Dirac operator introduced in [8].…”