Quantum isometries and noncommutative spheres
The athors build noncommutative analogs of the usual spheres \(S^{n-1}\subset{\mathbb R}^n\), by following a fashionable approach that aims at considering noncommutative geometric objects, defined by some noncommutative algebras that reduce, in the commutative case, to algebras having as spectra commutative manifolds. So, they axiomatize noncommutative spheres by two noncommutative algebras, denoted \(A^*_n\) and \(A^+_n\), respectively, that in the commutative case reduce to \(A_n=C^*\left(x_1,\dots, x_n \mid x_j=x_j^*, \;x_jx_i=x_ix_j,\;\sum x_i^2=1\right)\), whose spectrum is just \(S^{n-1}\subset{\mathbb R}^{n}\). Furthermore they study the ``differential structure'' of such noncommutative spheres by recognizing there spectral triples in the sense of A. Connes. Then, they obtain ``quantum isometries'' of such noncommutative spheres by using some previous works where quantum isometries are defined in the framework of spectral triples. The paper, after a concise introduction, splits into eight more sections. 1. The usual sphere. 2. Noncommutative spheres. 3. Projective spaces. 4. Probabilistic aspects. 5. Spherical integrals. 6. Spectral triplets. 7. Quantum isometries. 8. Concluding remarks. Remark. At the end of section 7, the authors recognize that ``There are several questions raised by the above results, concerning the axiomatization of the noncommutative spheres. Perhaps the most important one is the following: What conditions on a spherical algebra \(A\) ensure the fact that the corresponding quantum isometry group is ``easy'' in the sense of [6]?''. Really a more natural way to consider noncommutative differential geometry is one in the framework of pseudogroups, as adopted by the reviewer of this paper, in order to consider quantum (super)manifolds and quantum (super) PDEs. See also the review to the paper by \textit{J. Bhowmick} and \textit{D. Goswami} [J. Funct. Anal. 257 No.~8, 2530--2572 (2009; Zbl 1180.58005)]. See also the following paper: \textit{A. Prástaro} [Nonlinear Anal., Theory Methods Appl., Ser. A, theory Methods 71, No.~12, e502--e525 (2009), \url{doi:10.1016/j.na.2008.11.077}], where quantum spheres are considered in a natural way to obtain useful tools to formulate an algebraic topology for quantum (super)manifolds.
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