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].
讲师
日期
2025年03月24日 至 07月07日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一 | 13:30 - 16:55 | A3-4-301 | ZOOM 03 | 242 742 6089 | BIMSA |
参考资料
[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.
听众
Graduate
, Advanced Undergraduate
视频公开
不公开
笔记公开
不公开
语言
中文
讲师介绍
王亦龙于2018年从俄亥俄州立大学数学专业博士毕业，之后在路易斯安那州立大学任博士后，并于2021年加入BIMSA任助理研究员。主要研究方向为量子代数与量子拓扑，具体课题包括模张量范畴及其对应的拓扑量子场论的代数与数论性质。