Introduction to Quantum Groups - NMAG589

• Title: Introduction to Quantum Groups
• Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2025
• Semester: summer
• E-Credits: 3
• Hours per week, examination: summer s.:2/0, Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: taught
• Language: English
• Teaching methods: full-time
• Guarantor: Dr. Re O'Buachalla, Dr.
• Teacher(s): Dr. Re O'Buachalla, Dr.
• Class: M Mgr. MSTR; M Mgr. MSTR > Volitelné
• Classification: Mathematics > Algebra

Annotation

English

This course covers key topics in quantum group theory, in particular for the theory of Drinfeld-Jimbo quantum groups. Topics covered include Hopf algebras, quantised enveloping algebras, the classification of finite- dimensional modules, quantum Verma modules, multiplicity formulae, Lusztig's PBW Theorem, the dual picture of general quantised coordinate algebras, Yang-Baxter solutions, the FRT construction, coquastriangular structures, and quantum homogeneous spaces.

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)

Czech

Kurz pokrývá klíčová témata teorie kvantových grup, zejména teorii Drinfeldových-Jimbových kvantových grup. Zahrnuje témata Hopfových algeber, kvantovaných obalovacích algeber, klasifikaci konečnědimenzionálních modulů, kvantové Vermovy moduly, formule pro násobnosti, Lusztigovu PBW větu, duál obecných kvantovaný souřadnicových algeber, řešení Yangovy-Baxterovy rovnice, FRT konstrukci, kokvazitriangulární struktury a kvantové homogenní prostory.

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)

Course completion requirements

English

Submission of the project.

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)

Czech

Připravení krátkého projektu na téma související s přednáškou.

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)

Literature

English

Humphreys, J. E. (1972). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics

Christian Kassel (1995). Quantum Groups

Klimyk, A., & Schmüdgen, K. (1997). Quantum Groups and Their Representations

Lusztig, G. (1994). Introduction to Quantum Groups

Majid, S. (2002). A Quantum Groups Primer

Nesheveyev, S., & Tuset, L. (2013). Compact Quantum Groups and Their Representation Categories

Yuncken, R., & Voigt, C. (2020). Complex Semisimple Quantum Groups and Representation Theory

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)

Syllabus

English

Hopf Algebras & Basic Examples: Definition of Hopf algebras, dual Hopf algebras, low-dimensional cases, algebraic groups, and universal enveloping algebras The dual pair Uq(sl2) and Oq(SU2) Modules and Comodules: Representation theory for Uq(sl2) and Oq(SU2), including the Peter-Weyl decomposition. Verma Modules & Multiplicity Formulas: The quantum category O and the study of Verma modules and their multiplicities. Complex semisimple Lie Algebra and their universal enveloping algebra: Presentation of universal algebras using Serre relations. Drinfeld-Jimbo Quantum Groups: Overview of Drinfeld-Jimbo quantum groups and their representation theory. Finite-Dimensional Representations: Classification of finite-dimensional representations, Verma module construction, and worked examples of multiplicity formulas. Lusztig’s PBW Theorem: Proof of Lusztig’s PBW theorem with examples, including applications. Oq(G): The FRT construction, dual pairing in the A-series, and coquasi-triangular structure, and the braiding on the category of comodules. Quantum homogeneous spaces: General facts, classical examples, and quantum flag manifolds.

The student will also be expected to prepare a short project on more advanced topics in quantum groups, such as canonical bases, crystal bases, quantum symmetric spaces, or noncommutative differential geometry of quantum homogeneous spaces.

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)

Czech

1) Hopf Algebras & Basic Examples: Definition of Hopf algebras, dual Hopf algebras, low-dimensional cases, algebraic groups, and universal enveloping algebras . The dual pair Uq(sl2) and Oq(SU2)

2) Modules and Comodules: Representation theory for Uq(sl2) and Oq(SU2), including the Peter-Weyl decomposition.

3) Verma Modules & Multiplicity Formulas: The quantum category O and the study of Verma modules and their multiplicities.

4) Complex semisimple Lie Algebra and their universal enveloping algebra: Presentation of universal algebras using Serre relations.

5) Drinfeld-Jimbo Quantum Groups: Overview of Drinfeld-Jimbo quantum groups and their representation theory. Finite-Dimensional Representations: Classification of finite-dimensional representations, Verma module construction, and worked examples of multiplicity formulas.

6) Lusztig’s PBW Theorem: Proof of Lusztig’s PBW theorem with examples, including applications.

7) Oq(G): The FRT construction, dual pairing in the A-series, and coquasi-triangular structure, and the braiding on the category of comodules.

8) Quantum homogeneous spaces: General facts, classical examples, and quantum flag manifolds.

The student will also be expected to prepare a short project on more advanced topics in quantum groups, such as canonical bases, crystal bases, quantum symmetric spaces, or noncommutative differential geometry of quantum homogeneous spaces.

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)

Entry requirements

English

Knowledge of basics of modern algebra and Lie groups.

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)

Czech

Základy moderní algebry a Lieových grup.

Last update: Šmíd Dalibor, Mgr., Ph.D. (04.07.2025)