Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- Introduction
- 1 Auslander–Reiten Theory of Finite-Dimensional Algebras
- 2 τ-tilting Theory – an Introduction
- 3 From Frieze Patterns to Cluster Categories
- 4 Infinite-dimensional Representations of Algebras
- 5 The Springer Correspondence
- 6 An Introduction to Diagrammatic Soergel Bimodules
- 7 A Companion to Quantum Groups
- 8 Infinite-dimensional Lie Algebras and Their Multivariable Generalizations
- 9 An Introduction to Crowns in Finite Groups
- 10 An Introduction to Totally Disconnected Locally Compact Groups and Their Finiteness Conditions
- 11 Locally Analytic Representations of p-adic Groups
- References
7 - A Companion to Quantum Groups
Published online by Cambridge University Press: 25 November 2023
Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- Introduction
- 1 Auslander–Reiten Theory of Finite-Dimensional Algebras
- 2 τ-tilting Theory – an Introduction
- 3 From Frieze Patterns to Cluster Categories
- 4 Infinite-dimensional Representations of Algebras
- 5 The Springer Correspondence
- 6 An Introduction to Diagrammatic Soergel Bimodules
- 7 A Companion to Quantum Groups
- 8 Infinite-dimensional Lie Algebras and Their Multivariable Generalizations
- 9 An Introduction to Crowns in Finite Groups
- 10 An Introduction to Totally Disconnected Locally Compact Groups and Their Finiteness Conditions
- 11 Locally Analytic Representations of p-adic Groups
- References
Summary
In this chapter we give an introductory account of quantum groups (and more generally, quasitriangular bialgebras), allowing the reader to familiarize themselves with the basic concepts and constructions in this modern area of study in algebra and representation theory. This material accompanies online graduate lectures on quantum groups given by the author at the London Mathematical Society Autumn Algebra School in 2020. We also provide some supplementary material, mainly in the last section on cylindrical quasitriangularity.
- Type
- Chapter
- Information
- Modern Trends in Algebra and Representation Theory , pp. 227 - 272Publisher: Cambridge University PressPrint publication year: 2023
References
[1]Appel, A., and Vlaar, B. 2020. Universal k-matrices for quantum Kac-Moody algebras. arXiv preprint arXiv:2007.09218.Google Scholar
[2]Balagovic´, M., and Kolb, S. 2019. Universal K-matrix for quantum symmetric pairs. Journal fu¨r die reine und angewandte Mathematik, 2019(747), 299–353.Google Scholar
[3]Bao, H, and Wang, W. 2018. A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs. Socie´te´ mathe´matique de France Paris.Google Scholar
[4]Bernstein, I N, Gelfand, I M, and Gelfand, S I. 1975. Differential operators on the base affine space and a study of g-modules. Lie groups and their representations (Proc. Summer School, Bolyai Ja´nos Math. Soc., Budapest, 1971), 21–64.Google Scholar
[5]Bourbaki, N. 1994. Lie groups and Lie algebras. Pages 247–267 of: Elements of the History of Mathematics. Springer.Google Scholar
[6]Chari, V, and Pressley, A. 1991. Quantum affine algebras. Communications in Mathematical Physics, 142(2), 261–283.Google Scholar
[7]Chari, V, and Pressley, A. 1995. A guide to quantum groups. Cambridge University Press.Google Scholar
[8]Cherednik, I. 1984. Factorizing particles on a half-line and root systems. Theoretical and Mathematical Physics, 61(1), 977–983.Google Scholar
[9]Cherednik, I. 1992. Quantum Knizhnik-Zamolodchikov equations and affine root systems. Communications in Mathematical Physics, 150(1), 109–136.Google Scholar
[10]Dascalescu, S, Nastasescu, C, and Raianu, S. 2001. Hopf Algebras: an Introduction. Marcel Decker Inc., New York.CrossRefGoogle Scholar
[11]Dobson, L, and Kolb, S. 2019. Factorisation of quasi K-matrices for quantum symmetric pairs. Selecta Mathematica, 25(4), 1–55.Google Scholar
[12]Drinfeld, V G. 1985. Hopf algebras and the quantum Yang-Baxter equation. Soviet Math. Dokl., 32(6), 254–258.Google Scholar
[13]Drinfeld, V G. 1987. Quantum Groups. Pages 798–820 of: Gleason, A M (ed), Proceedings of the International Congress of Mathematicians. American Mathematical Society.Google Scholar
[14]Ehrig, M, and Stroppel, C. 2018. Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality. Advances in Mathematics, 331, 58–142.Google Scholar
[15]Frenkel, I B, and Reshetikhin, N Yu. 1992. Quantum affine algebras and holonomic difference equations. Communications in Mathematical Physics, 146(1), 1–60.Google Scholar
[16]Ha¨ring-Oldenburg, R. 2001. Actions of tensor categories, cylinder braids and their Kauffman polynomial. Topology and its Applications, 112(3), 297–314.Google Scholar
[17]Jantzen, J C. 1996. Lectures on quantum groups. Vol. 6. American Mathematical Soc.Google Scholar
[18]Jimbo, M. 1986. A q-analogue of U (gl(N + 1)), Hecke algebra, and the Yang-Baxter equation. Letters in Mathematical Physics, 11(3), 247–252.Google Scholar
[19]Jimbo, M, and Miwa, T. 1994. Algebraic analysis of solvable lattice models. Vol. 85. Amer. Math. Soc.Google Scholar
[20]Joyal, A, and Street, R. 1993. Braided tensor categories. Advances in Mathematics, 102(1), 20–78.Google Scholar
[22]Kamnitzer, J, and Tingley, P. 2009. The crystal commutor and Drinfeld’s unitarized R-matrix. Journal of Algebraic Combinatorics, 29(3), 315–335.Google Scholar
[24]Klimyk, A, and Schmu¨dgen, K. 1997. Quantum Groups and their Representations (1997). Springer Texts and Monographs in Physics.CrossRefGoogle Scholar
[25]Kolb, S. 2020. Braided module categories via quantum symmetric pairs. Proceedings of the London Mathematical Society, 121(1), 1–31.Google Scholar
[26]Kostant, B. 1977. Graded manifolds, graded Lie theory, and prequantization. Pages 177–306 of: Differential Geometrical Methods in Mathematical Physics. Springer.Google Scholar
[27]Kulish, P P, and Reshetikhin, N Yu. 1983. Quantum linear problem for the sine-Gordon equation and higher representations. Journal of Soviet Mathematics, 23(4), 2435–2441.Google Scholar
[28]Kulish, P P, Sasaki, R, and Schwiebert, C. 1993. Constant solutions of reflection equations and quantum groups. Journal of Mathematical Physics, 34(1), 286–304.Google Scholar
[29]Letzter, G. 1999. Symmetric pairs for quantized enveloping algebras. Journal of Algebra, 220(2), 729–767.Google Scholar
[31]Majid, S. 1988. Hopf algebras for physics at the Planck scale. Classical and Quantum Gravity, 5(12), 1587.Google Scholar
[33]Noumi, M, and Sugitani, T. 1995. Quantum symmetric spaces and related q-orthogonal polynomials. Pages 28–40 of: Arima, A et al. (ed), Group theoretical methods in physics. World Scientific.Google Scholar
[34]Noumi, M, Dijkhuizen, M S, and Sugitani, T. 1997. Multivariable Askey-Wilson polynomials and quantum complex Grassmannians. AMS Field Inst. Commun., 14, 167–177.Google Scholar
[35]Reshetikhin, N. 1990. Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Letters in Mathematical Physics, 20(4), 331–335.Google Scholar
[36]Reshetikhin, N. 2010. Lectures on the integrability of the six-vertex model. Exact methods in low-dimensional statistical physics and quantum computing, 197–266.Google Scholar
[37]Reshetikhin, N Yu, and Turaev, V G. 1990. Ribbon graphs and their invariants derived from quantum groups. Communications in Mathematical Physics, 127(1), 1–26.Google Scholar
[38]Saito, Y. 1994. PBW basis of quantized universal enveloping algebras. Publications of the Research Institute for Mathematical Sciences, 30(2), 209–232.Google Scholar
[40]Sklyanin, E K. 1988. Boundary conditions for integrable quantum systems. Journal of Physics A: Mathematical and General, 21(10), 2375.Google Scholar
[42]Tanisaki, T. 1992. Killing forms, Harish-Chandra isomorphisms, and universal R-matrices for quantum algebras. International Journal of Modern Physics A, 7(supp01b), 941–961.Google Scholar
[43]tom Dieck, T, and Ha¨ring-Oldenburg, R. 1998. Quantum groups and cylinder braiding. Pages 619–639 of: Forum Math., vol. 10.Google Scholar