1Universitat Zürich, Zürich, Switzerland. Email: camiloariasabad@gmail.com
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.
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.
Texto completo disponible en PDF
References
[1] J. Alexander, `A Lemma on a System of Knotted Curves´, Proc. Nat. Acad. Sci. 9, (1923), 93-95.
[2] V. Arnold, `The Cohomology Ring of the Group of Dyed Braids´, Mat. Zametki 5, (1970), 227-231.
[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).
[5] J. Birman and T. Brendle, Braids: A Survey, Handbook of Knot Theory, Elsevier, 2005.
[6] W. Burau, `Über Zopfgruppen Und Gleischsinning Verdrillte Verkettungen´, Abh. Math. Sem. Ham. II, (1936), 171-178.
[7] V. Drinfeld, `Quantum Groups´, Proceedings of the 1986 ICM, (1986), 798-820. Amer. Math, Soc.
[8] V. Drinfeld, `Quasi-Bialgebras´, Leningrad Math. Journal 1, (1990), 1419-1457.
[9] E. Fadell and J. V. Buskirk, `The Braid Groups of E2 and S2´, Duke Mathematical Journal 29, (1962), 243-257.
[10] B. Farb and D. Margalit, A Primer on the Mapping Class Group, Princeton University Press, 2011.
[11] R. Fox and L. Neuwirth, `The Braid Groups´, Math. Scand. 10, (1962), 119-126.
[12] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, `Graduate texts in mathematics´, 1972, Springer-Verlag.
[13] M. Jimbo, `A q-Difference Analogue of \mathsfU(\mathfrakg) and the Yang-Baxter Equation´, Letters in Math. Phys. 10, (1985), 63-69.
[14] C. Kassel, Quantum Groups, Springer Verlag, 1994.
[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.
[19] D. Krammer, `Braid Groups Are Linear´, Ann. Math. 155, 2 (2002), 131-156.
[20] R. J. Lawrence, `Homological Representations of the Hecke Algebra´, Comm. Math. Phys. 135, 1990 (0000), 141-191.
[21] A. Markov, `Über die freie äquivalenz geschlossener zöpfe´, Recueil Math. Moscou 1, (1935).
[22] D. Sinha, The Homology of the Little Disks Operad, `Arxiv:math0610236v3´, (0000).
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}
}