(in Polish) Kwantowa Teoria Galois

elective monographs

The course is essentially self-contained.

Only basic experience and familiarity with simplest properties of rings, modules, groups, topological spaces, covering spaces and fibrations.

No particular knowledge about Galois theory is assumed.

Classical Galois theory, whose original purpose was to give a necessary and sufficient condition for solvability in radicals of an algebraic equation, has been modified and widely generalised. The list of important examples of Galois/covering theories includes: theory of covering spaces and the fundamental group, which goes back to H. Poincaré and is often considered as the main origin of algebraic topology; étale Galois theory of schemes, due to A. Grothendieck, which plays an important role in algebraic geometry and has important applications in algebraic number theory.

This course will describe the current stage of the generalisation process encompassing noncommutative algebras. Among them are Hopf-Galois extensions and principal actions of quantum groups on C*-algebras (noncommutative principal bundles). Finally, the possible applications to quantum invariants of manifolds will be discussed.

First, an overview of structures appearing in geometrically significant ramifications of the classical Galois theory: Galois field extensions; Galois theory as equivalence of categories; fundamental group and universal covering; torsors and Galois cohomology; Galois context and principal fibrations; Tannakian categories and Tannakian reconstruction.

Next, the (noncommutative) Hopf-Galois theory will be overviewed: Hopf-Galois extensions: crossed products as cleft extensions, Schneider's theorems; differential-geometric aspects of Hopf-Galois theory (strong connections, principal comodule algebras, free actions of compact quantum groups on unital C*-algebras); the Chern-Galois character (entwining structures, principal extensions, associated modules, index pairings); corings (the Sweedler coring and the coring associated to an entwining structure, separable and split algebra extensions, corings with a grouplike element and their relation to connections.

Finally, the Hopf-Galois separable extensions of commutative algebras will be applied for constructing a new quantum invariant of manifolds which enhances the fundamental group from the group level to the level of Hopf algebras. The comparison between our quantum invariant and the classical fundamental group will be given with use of the quantum Maurer–Cartan equation and the methods of Synthetic Differential Geometry.

The selected fragments of:

-- Steven H. Weintraub, Galois Theory, Springer Science & Business Media, 2008.

-- David E. Radford, Hopf Algebras, World Scientific, 2012.

-- Hans-Jürgen Schneider,Principal homogeneous spaces for arbitrary Hopf algebras, Isr. J. Math.72 (1990), 167–195.

-- Susan Montgomery, Hopf Galois theory: A survey, Geometry & Topology Monographs 16 (2009), 367–400

-- Peter Schauenburg, Hans-Jürgen Schneider, On generalized Hopf galois extensions, Journal of Pure and Applied Algebra 202 (2005) 168 – 194.

-- Paul F. Baum, Piotr M. Hajac, Local Proof of Algebraic Characterization of Free Actions, SIGMA, 2014, Volume 10, 060.

-- Brzeziński T., Hajac P.M., The Chern–Galois character, C. R. Math. Acad. Sci. Paris 338 (2004), 113–116, math.KT/0306436.

-- André Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, in Part II of Category Theory, Proceedings, Como 1990, eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lec. Notes in Mathematics 1488, Springer, Berlin, 1991, pp. 411–492.

-- Kock, A., Synthetic Differential Geometry, London Mathematical Society Lecture Note Series, vol. 51, Cambridge University Press, Cambridge 1981.