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Principal Fiber Bundles in Non-commutative Geometry

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Book cover Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

These are the expanded notes of a course given at the Summer school “Geometric, topological, and algebraic methods for quantum field theory” held at Villa de Leyva, Colombia, in July 2015. We first give an introduction to non-commutative geometry and to the language of Hopf algebras. We next build up a theory of non-commutative principal fiber bundles and consider various aspects of such objects. Finally, we illustrate the theory using the quantum enveloping algebra \(U_q\, {\mathfrak {s}}\text {l}(2)\) and related Hopf algebras.

Al álgebra le dediqué mis mejores ánimos,

no sólo por respeto a su estirpe clásica

sino por mi cariño y mi terror al maestro.

Gabriel García Márquez,

Vivir para contarla [21]

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Notes

  1. 1.

    That is, there exists a continuous map \(\Phi : X' \times [0,1] \rightarrow X\) such that \(\Phi (x,0) = \varphi _0(x)\) and \(\Phi (x,1) = \varphi _1(x)\) for all \(x\in X'\).

  2. 2.

    Recall that \(\delta _{x,y} = 1\) if \(x = y\) and \(\delta _{x,y} = 0\) otherwise.

  3. 3.

    Drinfeld along with other invited mathematicians from the Soviet Union was prevented by the Soviet authorities to attend the conference; in Drinfeld’s absence his contribution was read by Cartier.

  4. 4.

    The concept of enveloping algebra of a Lie algebra is a classical concept of the theory of Lie algebras; see for instance [15, 28, 31, 54]. The relationship between the quantum enveloping algebra \(U_q\, {\mathfrak {s}}\text {l}(2)\) and the enveloping algebra of the Lie algebra \({\mathfrak {s}}\text {l}(2)\) is explained in [31, VI.2].

  5. 5.

    The concept of the cotensor product of comodules was first introduced in [20]. See also [46, 58].

  6. 6.

    For this to hold we need the extra faithful flatness condition mentioned in Sect. 3.7.1, Remark 3.5.

  7. 7.

    The Krull dimension of \(\mathscr {B}_H\) is the dimension of the algebraic variety V such that \(\mathscr {B}_H = \mathscr {O}(V)\).

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Acknowledgements

I thank the organizers of the Summer school “Geometric, topological and algebraic methods for quantum field theory” held at Villa de Leyva, Colombia, in July 2015 for inviting me to give the course that led to these notes. I am also grateful to the students for their feedback and to Sylvie Paycha for her careful reading of these notes and her suggestions. Finally let me mention the travel support I received from the Institut de Recherche Mathématique Avancée in Strasbourg.

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  1. Institut de Recherche Mathématique Avancée, CNRS & Université de Strasbourg, 7 rue René Descartes, 67084, Strasbourg, France

    Christian Kassel

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Editors and Affiliations

  1. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia

    Alexander Cardona

  2. Department of Mathematics, The University of Texas, Austin, Texas, USA

    Pedro Morales

  3. Departamento de Física, Universidad del Valle, Cali, Colombia

    Hernán Ocampo

  4. Institut für Mathematik, Universität Potsdam, Potsdam, Germany

    Sylvie Paycha

  5. Departamento de Física, Universidad de los Andes, Bogotá, Colombia

    Andrés F. Reyes Lega

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Kassel, C. (2017). Principal Fiber Bundles in Non-commutative Geometry. In: Cardona, A., Morales, P., Ocampo, H., Paycha, S., Reyes Lega, A. (eds) Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-65427-0_3

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