Introducción a las representaciones de los grupos de trenzas

Introduction to Representations of Braid Groups

Estas notas fueron preparadas para un minicurso enseñado en la
escuela Cimpa Algebraic and geometric aspects of representation theory, en
Curitiba, Brazil en Marzo de 2013. El propósito del curso es presentar una
introducción al estudio de las representaciones de los grupos de trenzas. Tres
clases generales de representaciones son consideradas: representaciones homol
ógicas de mapping class groups, representaciones de monodromía de la
connección de Knizhnik-Zamolodchikov, y soluciones de la equación de Yang-
Baxter en términos de quasi-triangular bialgebras. Algunas de las notables
relaciones entre estas construcciones son descritas.

These are lecture notes prepared for a minicourse given at the
Cimpa Research School Algebraic and geometric aspects of representation the-
ory, held in Curitiba, Brazil in March 2013. The purpose of the course is to
provide an introduction to the study of representations of braid groups. Three
general classes of representations of braid groups are considered: homological
representations via mapping class groups, monodromy representations via the
Knizhnik-Zamolodchikov connection, and solutions of the Yang-Baxter equation
via quasi-triangular bialgebras. Some of the remarkable relations between
these three dierent constructions are described.

Introduction to Representations of Braid Groups Introducción a las representaciones de los grupos de trenzas CAMILO ARIAS ABAD¹

¹Universitat Zürich, Zürich, Switzerland. Email: camiloariasabad@gmail.com

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Abstract

These are lecture notes prepared for a minicourse given at the Cimpa Research School Algebraic and geometric aspects of representation theory, held in Curitiba, Brazil in March 2013. The purpose of the course is to provide an introduction to the study of representations of braid groups. Three general classes of representations of braid groups are considered: homological representations viamapping class groups, monodromy representations via the Knizhnik-Zamolodchikov connection, and solutions of the Yang-Baxter equation via quasi-triangular bialgebras. Some of the remarkable relations between these three different constructions are described.

Key words: Braid groups, Representation theory.

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2000 Mathematics Subject Classification: 20F36, 06B15.

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Resumen

Estas notas fueron preparadas para un minicurso enseñado en la escuela Cimpa Algebraic and geometric aspects of representation theory, en Curitiba, Brazil en Marzo de 2013. El propósito del curso es presentar una introducción al estudio de las representaciones de los grupos de trenzas. Tres clases generales de representaciones son consideradas: representaciones homológicas de mapping class groups, representaciones de monodromía de la connección de Knizhnik-Zamolodchikov, y soluciones de la equación de Yang-Baxter en términos de quasi-triangular bialgebras. Algunas de las notables relaciones entre estas construcciones son descritas.

Palabras clave: Grupos de trenzas, teoría de representaciones.

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Texto completo disponible en PDF

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References

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[3] E. Artin, `Theory of Braids´, Ann. of Math. 48, 2 (1947), 101-126.

[4] S. Bigelow, `Representations of Braid Groups´, Proceedings ICM Vol II, 2002 (37--45).

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[10] B. Farb and D. Margalit, A Primer on the Mapping Class Group, Princeton University Press, 2011.

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[15] C. Kassel and V. Turaev, Braid Groups, Springer Verlag, 2008.

[16] T. Kohno, Conformal Field Theory and Topology, `Translations of Math. Monographs´, (0000), Amer. Math. Soc..

[17] T. Kohno, `Monodromy Representations of Braid Groups and Yang-Baxter Equations´, Ann. Inst. Fourier 37, (1987), 139-160.

[18] T. Kohno, `Quantum and Homological Representations of Braid Groups, Configuration Spaces, Geometry, Combinatorics and Topology´, Edizioni della Normale, (2012), 355-372.

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[20] R. J. Lawrence, `Homological Representations of the Hecke Algebra´, Comm. Math. Phys. 135, 1990 (0000), 141-191.

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(Recibido en mayo de 2013. Aceptado en enero de 2015)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv49n1a01,
    AUTHOR  = {Arias Abad, Camilo},
    TITLE   = {{Introduction to Representations of Braid Groups}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2015},
    volume  = {49},
    number  = {1},
    pages   = {1--38}
}

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