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. |
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. |
|
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). |
|
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. Structure of Hopf algebras: Cartier-Milnor-Moore theorem. |
|
2 May |
Lecture 12. Structure of Hopf algebras: Poincaré–Birkhoff–Witt theorem. |
|
7 May |
Lecture 13. Characters and infinitesimal characters. |
|
|
|
|
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
Final projects
To be announced.