Geometry Elective 1 - NMAG496 (Hopf algebras and Hopf-Galois theory)

• Title: Výběrová přednáška z geometrie 1
• Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2024
• Semester: winter
• E-Credits: 3
• Hours per week, examination: winter 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: Thomas Weber, Ph.D.
• Teacher(s): Thomas Weber, Ph.D.
• Class: M Mgr. MSTR; M Mgr. MSTR > Volitelné
• Classification: Mathematics > Geometry

Annotation

English

In this course we discuss the fundamentals of Hopf algebras, the noncommutative generalization of groups and Lie algebras. The algebraic concepts are introduced together with their categorical counterparts: Hopf algebras correspond to monoidal structures on their representation categories. A particular focus is given on examples, including matrix quantum groups, which play a prominent role in mathematical physics. Afterwards, we study the concept of Hopf-Galois extension, which generalizes Galois extension of fields and which can be interpreted as quantum principal bundle.

Last update: Šmíd Dalibor, Mgr., Ph.D. (12.09.2024)

Czech

V kurzu se budeme zabývat Hopfovými algebrami, které jsou nekomutativním zobecněním grup a Lieových algeber. V teorii kategorií korespondují Hopfovy algebry monoidálním strukturám na svých reprezentačních kategoriích. Zaměříme se na příklady zahrnující i maticové kvantové grupy, které hrají prominentní roli v matematické fyzice. Následně zavedeme pojem Hopfova-Galoisova rozšíření, který zobecňuje Galoisovo rozšíření těles a dá se interpertovat jako kvantový hlavní bandl.

Last update: Šmíd Dalibor, Mgr., Ph.D. (12.09.2024)

Aim of the course

The goal of the course is to study Hopf algebras from an algebraic and categorical point of view. We are going to internalize the axioms of Hopf algebra theory via numerous explicit examples, such as q-deformed SL(2). The participants will become acquainted with Sweedler's coproduct notation, which will enable them to compactly perform computations in the Hopf algebra formalism. The notion of Hopf-Glaois extension will be discussed and we will explore its role in noncommutative geometry as quantum principal bundle.

Last update: Weber Thomas, Ph.D. (12.09.2024)

Course completion requirements

The final exam consists of a seminar talk of about 45 minutes presented by the participant. For this, a number of possible seminar topics will be given at the end of the course. The material will be based on the content of the course but is meant to extend the discussed material. For interested students there might be the possibility to continue the course project in form of a master thesis.

Last update: Weber Thomas, Ph.D. (12.09.2024)

Literature

Hopf algebras:

Chapter III, IV and XI of

C. Kassel, Quantum groups. Graduate Texts in Mathematics, 155. Springer-Verlag, New York, 1995.

Hopf-Galois extensions:

S. Montgomery, Hopf Galois theory: A survey, Geometry & Topology Monographs 16 (2009) 367–400.

Last update: Weber Thomas, Ph.D. (12.09.2024)

Teaching methods

Blackboard class with seminars at the end of the course.

Last update: Weber Thomas, Ph.D. (12.09.2024)

Syllabus

.

Last update: Weber Thomas, Ph.D. (12.09.2024)