Published October 13, 2023 | Version v1
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Introduction to quantum groups

Creators

  • 1. Stellenbosch University

Description

Brief introductory seminar series presented to my research group (led by Bruce Bartlett), with a few guests.

Our goal is to introduce quantum groups assuming little background, showing a few examples, but focusing mainly on the perspective that quantum groups give rise to link invariants through their representation category.

The presentation is far from exhaustive - this topic is vast, and a thorough treatment requires little short of an entire textbook. That said, my notes include references (in pink) for each section, which should help an interested reader dive deeper. Notable omissions are the Drinfeld quantum double, the construction of quantum sl(2), and proofs of several assertions.

Breakdown of talks

  1. 29/09/2023
    We motivate the Yang-Baxter equation, introduce universal enveloping algebras & Hopf algebras, & see our first examples of quantum groups
  2. 06/10/2023
    We see another example of a quantum group, show how braided Hopf algebras give Yang-Baxter equation solutions, & discuss the square of the antipode map in a Hopf algebra; finally, we define ribbon categories
  3. 13/10/2023
    We revise graphical calculi for various kinds of category, connect various kinds of Hopf algebra (braided, ribbon, etc) with their categories of representations, & finally show how ribbon Hopf algebras produce the Reshetikhin-Turaev link invariant

Files

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Additional details

Dates

Created
2023-09-29/2023-10-13

References

  • Kassel, C. (1995). Quantum Groups (Vol. 155). Springer New York. https://doi.org/10.1007/978-1-4612-0783-2
  • Selinger, P. (2010). A Survey of Graphical Languages for Monoidal Categories. In B. Coecke (Ed.), New Structures for Physics (Vol. 813, pp. 289–355). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-12821-9_4
  • Bakalov, B., & Kirillov, A. A. (2001). Lectures on tensor categories and modular functors. American Mathematical Society.
  • Gelca, R. (2014). Theta Functions and Knots. World Scientific. https://doi.org/10.1142/8872
  • Heunen, C. J. M., Sadrzadeh, M., & Grefenstette, E. (Eds.). (2013). Quantum physics and linguistics: A compositional, diagrammatic discourse (First edition). Oxford University Press.
  • Turaev, V. G. (2016). Quantum Invariants of Knots and 3-Manifolds. De Gruyter. https://doi.org/10.1515/9783110435221
  • Street, R. (2007). Quantum groups: A path to current algebra. Cambridge University Press.
  • Joyal, A., & Street, R. (1993). Braided Tensor Categories. Advances in Mathematics, 102(1), 20–78. https://doi.org/10.1006/aima.1993.1055
  • Helgason, S. (1979). Differential geometry, Lie groups, and symmetric spaces. Academic press.
  • Hilgert, J., & Neeb, K.-H. (2012). Structure and Geometry of Lie Groups. Springer New York. https://doi.org/10.1007/978-0-387-84794-8
  • Perk, J. H., & Au-Yang, H. (2006). Yang-baxter equations. arXiv preprint math-ph/0606053.