Hopf Algebras in Combinatorics

TU Graz - Summer semester 2024

Instructors: Adrián Celestino and Yannic Vargas
Email: celestino(at)math.tugraz.at and yvargaslozada(at)tugraz.at

Times and location: Tuesdays 10:15 – 12:00. Place: AE06 (STEG050).

Thursdays 10:15 – 12:00. Place: Seminarraum A306 (ST03014).

News:

● 16.04.2024. Updated notes Lecture Notes 1-6.
● 29.03.2024. Exercise Set 2 is now available.
● 19.03.2024. A draft of my notes of the first four lectures is available here. It may contain several typos. Feel free to contact me if you have any questions.
● 11.03.2024. Exercise Set 1 is now available.
● 01.03.2024. The lecture times habe been set.
● 31.01.2024. The initial meeting will be held on 1 March at 14:00. Location: Seminarraum A306 (ST03014), Steyrergasse 30/III. We will decide the lecture times.
If you cannot join, please send an email to Adrián or Yannic with your available/proposed times.
● 27.01.2024. The course website has been created.
● 27.01.2024. Testing HTML.


Course description

The objective of the course is to study the classical notion of Hopf algebras and relevant applications in Combinatorics. Roughly speaking, a Hopf algebra is a vector space together with two operations, called multiplication and comultiplication which are related in an interesting way. In combinatorics, the comultiplication operation models the notion of breaking down a complex structure into simpler components, mirroring the combinatorial principle of decomposing a counting problem into smaller, more manageable subproblems. The algebraic properties of Hopf algebras have proven to be useful in understanding and solving combinatorial problems, providing a powerful framework for studying the interplay between algebra and combinatorics.

A list of contents of the topics of the course can be found here.

Lecture plan

Date

Content

Remarks

1 Mar

Lecture 0. Initial meeting and course presentation

Lectures times have been set.
Tuesday and Thursday. 10:15 to 12:00.

5 Mar

Lecture 1. Tensor product of vector spaces


7 Mar

Lecture 2. Properties of tensor products. Algebras: definition and basic properties

The lecture will be held at Seminarraum A206 (ST02014).

12 Mar

Lecture 3. Coalgebras: definition and basic properties

Exercise Set 1 is now available.

14 Mar

Lecture 4. Isomorphism Theorem for coalgebras. Fundamental Theorem of Coalgebras.

Lecture Notes 1-4

19 Mar

Exercise lecture 1.


21 Mar

Lecture 5. Coproducts on the tensor algebra. Bialgebras: definition and basic properties.

Exercise Set 2 is now available.

9 Apr

Lecture 6. Examples of bialgebras. Primitive elements and Lie algebras. Convolution algebra Hom(C,A).

Lecture Notes 1-6

11 Apr

Lecture 7. Hopf algebras: defintion, basic properties and examples.


16 Apr

Exercise lecture 2.


18 Apr

Lecture 8. Hopf algebras. Properties of the antipode. Takeuchi formula for the antipode.


23 Apr

Lecture 9. Connected and graded bialgebras.


25 Apr

Lecure 10. Properties of graded connected bialgebras.


30 Apr

Lecture 11. Properties of graded connected bialgebras.


2 May

Lecture 12. Properties of graded connected bialgebras.


7 May

Lecture 13. Structure of Hopf algebras: Cartier-Milnor-Moore theorem.


23 May

Lecture 14. Structure of Hopf algebras: Poincaré-Birkhoff-Witt theorem.


28 May

Lecture 15. Characters and infinitesimal characters.


4 Jun

Lecture 16.







Exercise Sets


Grading

There will be exercise sessions during the course. Participation in these exercise sessions will count toward the grading. In addition, the students will be required to work on a final project on a selected topic related to the course together with an oral presentation about it.

Literature

We will not follow a textbook for the course. However, the lectures will follow parts of:

● David E. Radford. Hopf algebras. Vol. 49. World Scientific, 2011.

● Pierre Cartier and Frédéric Patras. Classical Hopf algebras and their applications. Vol. 2. Berlin Heidelberg: Springer, 2021.

Some secondary references and papers that will be relevant to the course:

● Moss E. Sweedler. Hopf algebras. Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.

● Dominique Manchon. Hopf algebras in renormalisation. Handbook of algebra, 5, 365-427, 2008

● Marcelo Aguiar and Swapneel Mahajan. Monoidal functors, species and Hopf algebras. Vol. 29. Providence, RI: American Mathematical Society, 2010.

● Darij Grinberg and Victor Reiner. Hopf algebras in combinatorics. arXiv preprint arXiv:1409.8356, 2014.

● Michiel Hazewinkel, Nadezhda Mikhailovna Gubareni, and Vladimir V. Kirichenko. Algebras, rings and modules: Lie algebras and Hopf algebras. Vol. 3. American Mathematical Soc., 2010