Principal bundles. The quantum case (Q2261586)

The book under review is a tentative to unitary collect several articles devoted to define the notion of principal bundles in noncommutative geometry, generically called ``quantum geometry''. This textbook, which addresses firstly to graduate students interested in mathematical physics, is the companion to the book [\textit{S. B. Sontz}, Principal bundles. The classical case. Cham: Springer (2015; Zbl 1321.53004)], which summarizes the notion of principal bundle in differential geometry, with many applications in theoretical and mathematical physics. The book under review contains a lot of interesting references, but firstly the author is very influenced by \textit{S.L. Woronowicz}'s approach [Lect. Notes Phys. 116, 407--412 (1980; Zbl 0513.46046)]. A good starting point for the reader should be the appendix B, dedicated to the Hopf algebras. The main goal of the book under review is the elaboration of Chapter 12, dedicated to quantum principal bundles, deeply influenced by the approach do to \textit{M. Đurđević} [Commun. Math. Phys. 175, No. 3, 457--520 (1996; Zbl 0840.58009)]. The previous chapters include preparatory notions needed to define the quantum principal bundles. Every chapter has a short introduction, where the author tries to justify the necessity to introduce the definition, underlying the differences between the definition of the ``classical'' object and the ``quantum'' one. The effort of the author is to find an ``intuitive'' motivation for the notion introduced. Each chapter ends with a Note, indicating the principal references used in the chapter. The notion of first-order differential calculus is reviewed in Chapter 2, as a generalization of the de Rham differential calculus, as was considered by \textit{S. L. Woronowicz} [Commun. Math. Phys. 122, No. 1, 125--170 (1989; Zbl 0751.58042)]. Essentially, a ``differential calculus'' over an algebra \(\mathcal{A }\) (generalizing the algebra of smooth functions) consists of an \(\mathcal{A}\)-bimodule \(\Gamma\) (generalizing the module of 1-forms), and a derivation \(d : \mathcal{A} \rightarrow \Gamma\) (corresponding to the exterior derivative \(d: C^{\infty}(M)\rightarrow \Gamma (T^*M))\). In Chapter 3 a Hopf algebra is endowed with a first-order differential calculus structure. Exemplification on the quantum function algebra \(\mathrm{SL}_q(2)\) and its dual pair \(\tilde{U}_q(sl_2)\) are presented. Next Chapters are devoted to adjoint co-action, covariant bimodules, braid groups and braided exterior algebra. Chapter 10 introduces higher-order differential calculi, following the quoted papers of Woronozicz's and Durdevich papers. The ``complex conjugation'' and the \(*\)-algebras are introduced in Chapter 11. Following \textit{M. Đurđević} [Rev. Math. Phys. 9, No. 5, 531--607 (1997; Zbl 0907.58001)], the quantum principal bundles are considered in Chapter 12 and endowed with differential calculus. Following the construction of principal bundles \(G\rightarrow P\rightarrow M\) in differential geometry, for quantum principal bundles are introduced the horizontals forms and the vertical algebra, the quantum connection and associated curvature, the quantum covariant derivative. The construction of quantum principale bundles is exemplified for: the trivial bundle, the quantum homogeneous bundles, and quantum Hopf bundles over Podles sphere. In the largely elaborated notes to Chapter 12, the author advocates to justify the approach chosen. Examples to motivate the general theory of the considered quantum principal bundle are considered in Chapter 13, devoted to the finite classical groups and the corresponding compact quantum groups, in the sense of \textit{S. L. Woronowicz} [Commun. Math. Phys. 111, 613--665 (1987; Zbl 0627.58034)], and Coxeter groups. The last example is connected with Dunkl operators as covariant derivatives in quantum principal bundle, based on a joint paper of \textit{M. Đurđevich} and \textit{S. B. Sontz} [SIGMA, Symmetry Integrability Geom. Methods Appl. 9, Paper 040, 29 p. (2013; Zbl 1282.81117)]. The paper ends with a short consideration of ``What next?''. The book contains a lot of exercises of different degree of difficulties, but the proofs of the theorems are done in detail. The book is very elaborate and very welcome not only for graduate students, but also for all people interested in mathematical physics.