Quantum Groups
&
Knot Invariants
PhD Course - Università degli studi di Firenze
Dates: 28 February- 26 April 2023
Hours: 20h - 4 CFU
Quantum groups and knot invariants
In mathematics and theoretical physics, the term ``quantum group'' denotes (different kinds of) non-commutative algebras with additional structure. These objects sit in the intersection between representation theory, low-dimensional topology, and quantum physics. This course is an introduction to quantum groups and monoidal categories, with an eye towards their connection with low-dimensional topology.
In particular, we will be interested in the description of Drinfel'd–Jimbo-type quantum groups (i.e. special kinds of Hopf algebras) and their application to the definition of knot invariants.
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The exam consists on a seminar of 1-1:30 h on one of the topics below and/or an oral exam covering the topics discussed in class.
Topics for the exam:
Drinfel'd doubles and Uq(slN)
References:
Chapters 3-4 of C. Kassel, M. Rosso, V. Turaev – Quantum Groups and Knot Invariants, Panoramas et Synthèses 5, Société Mathématique de France, Paris, 1997. vi+115 pp.
Quantum invariants of closed 3-manifolds
References:
Chapter 6 of C. Kassel, M. Rosso, V. Turaev – Quantum Groups and Knot Invariants, Panoramas et Synthèses 5, Société Mathématique de France, Paris, 1997. vi+115 pp.
Chapter II of V. Turaev – Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics, de Gruyter, Berlin-New York, 1994. 588 pp.
Chapter 8 of T. Ohtsuki – Quantum Invariants - a study of knots, 3-manifolds, and their sets, Series on Knots and Everything vol. 24, Worlds Scientific, Singapore-New Jersey-Hong Kong, 2002. xiii+ 489 pp.
Vassiliev invariants of links
References:
Chapter 8 of C. Kassel, M. Rosso, V. Turaev – Quantum Groups and Knot Invariants, Panoramas et Synthèses 5, Société Mathématique de France, Paris, 1997. vi+115 pp.
Chapter 7 of T. Ohtsuki – Quantum Invariants - a study of knots, 3-manifolds, and their sets, Series on Knots and Everything vol. 24, Worlds Scientific, Singapore-New Jersey-Hong Kong, 2002. xiii+ 489 pp.
Categorification
References:
M. Khovanov - A categorification of the Jones polynomial. Duke Mathematical Journal 101 , 2000, no. 3, pages 359–426, doi: 10.1215/S0012-7094-00-10131-7
P. Turner- Five lectures on Khovanov homology. Journal of Knot Theory and its Ramifications 26 (2017), no. 3, 1741009, 41 pp.
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A. Lauda - A categorification of quantum sl(2), Advances in Mathematics, Volume 225 (6), 2010, Pages 3327-3424, doi:10.1016/j.aim.2010.06.003.
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A. Beliakova, K. Habiro - A categorification of the ribbon element in quantum sl(2), Acta Mathematica Vietnamica 46, 2021, Pages 225–264, doi:10.1007/s40306-020-00399-7
Skein modules
References:
Chapters 2-3 of R. P. Bakshi - Skein Modules, Skein Algebras, and Their Ramifications, The George Washington University ProQuest Dissertations Publishing, 2021, 205 pp. ISBN: 979-8738-64577-8 (https://www.proquest.com/docview/2539900164)
TQFTs and their applications
References:
Chapter III of V. Turaev – Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics, de Gruyter, Berlin-New York, 1994. 588 pp.
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Chapter VI of V. Turaev – Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics, de Gruyter, Berlin-New York, 1994. 588 pp.
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P. Gustafson, M. S. Im, R. Kaldawy, M. Khovanov, Z. Lihn - Automata and one-dimensional TQFTs with defects, ArXiv:2301.00700
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L. Abrams - Two-dimensional topological quantum field theories and Frobenius algebras. J. Knot Theory Ramifications 5 (1996), no. 5, 569–587
J. Kock - Frobenius algebras and 2D topological quantum field theories (https://mat.uab.cat/~kock/TQFT/FS.pdf)