Algebraic structures in nonsemisimple TFTs
In this course, we will go over some of the basic algebraic structures that occur in current constructions of nonsemisimple topological field theories (TFTs). We will take a slow pace by starting with the representation theory of finite-dimensional associative algebras [3], and eventually cover the following topics:
1) Hopf algebras [8];
2) (Factorizable ribbon) finite tensor categories [4];
3) modified trace [6, 2];
4) pseudotrace [1];
5) Lyubashenko invariants and modular functor associated to factorizable ribbon finite tensor categories, in particular, the $\operatorname{SL}_2(\mathbb{Z})$ (projective) representations in this setting [7, 5].
1) Hopf algebras [8];
2) (Factorizable ribbon) finite tensor categories [4];
3) modified trace [6, 2];
4) pseudotrace [1];
5) Lyubashenko invariants and modular functor associated to factorizable ribbon finite tensor categories, in particular, the $\operatorname{SL}_2(\mathbb{Z})$ (projective) representations in this setting [7, 5].
Lecturer
Date
24th March ~ 7th July, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday | 13:30 - 16:55 | A3-4-301 | ZOOM 03 | 242 742 6089 | BIMSA |
Reference
[1] H. Bass. Euler characteristics and characters of discrete groups. Invent. Math., 35:155–196,
1976.
[2] A. Beliakova, C. Blanchet, and A. M. Gainutdinov. Modified trace is a symmetrised integral.
Selecta Math. (N.S.), 27(3):Paper No. 31, 51, 2021.
[3] C. W. Curtis and I. Reiner. Methods of representation theory. Vol. I. Pure and Applied Math-
ematics. John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and
orders, A Wiley-Interscience Publication.
[4] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor categories, volume 205 of Mathe-
matical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015.
[5] A. M. Gainutdinov and I. Runkel. Projective objects and the modified trace in factorisable
finite tensor categories. Compos. Math., 156(4):770–821, 2020.
[6] N. Geer, J. Kujawa, and B. Patureau-Mirand. M-traces in (non-unimodular) pivotal categories.
Algebr. Represent. Theory, 25(3):759–776, 2022.
[7] V. V. Lyubashenko. Invariants of 3-manifolds and projective representations of mapping class
groups via quantum groups at roots of unity. Comm. Math. Phys., 172(3):467–516, 1995.
[8] S. Montgomery. Hopf algebras and their actions on rings, volume 82 of CBMS Regional Confer-
ence Series in Mathematics. Published for the Conference Board of the Mathematical Sciences,
Washington, DC, 1993.
1976.
[2] A. Beliakova, C. Blanchet, and A. M. Gainutdinov. Modified trace is a symmetrised integral.
Selecta Math. (N.S.), 27(3):Paper No. 31, 51, 2021.
[3] C. W. Curtis and I. Reiner. Methods of representation theory. Vol. I. Pure and Applied Math-
ematics. John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and
orders, A Wiley-Interscience Publication.
[4] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor categories, volume 205 of Mathe-
matical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015.
[5] A. M. Gainutdinov and I. Runkel. Projective objects and the modified trace in factorisable
finite tensor categories. Compos. Math., 156(4):770–821, 2020.
[6] N. Geer, J. Kujawa, and B. Patureau-Mirand. M-traces in (non-unimodular) pivotal categories.
Algebr. Represent. Theory, 25(3):759–776, 2022.
[7] V. V. Lyubashenko. Invariants of 3-manifolds and projective representations of mapping class
groups via quantum groups at roots of unity. Comm. Math. Phys., 172(3):467–516, 1995.
[8] S. Montgomery. Hopf algebras and their actions on rings, volume 82 of CBMS Regional Confer-
ence Series in Mathematics. Published for the Conference Board of the Mathematical Sciences,
Washington, DC, 1993.
Audience
Graduate
, Advanced Undergraduate
Video Public
No
Notes Public
No
Language
Chinese
Lecturer Intro
Yilong Wang graduated from The Ohio State University in 2018. After working in Louisiana State University as a postdoc researcher, he joined BIMSA as an assistant research fellow in 2021. His research interests include modular tensor categories and topological quantum field theories.