Tensor Categories and their Modules

Seminar Description

Tensor categories are the categorical analogue of rings. They naturally arise when considering categories of objects which can be multiplied; e.g. representations of groups. They are ubiquitous in representation theory, and also play an important role in algebraic geometry, infinite dimensional Lie algebras, conformal field theory, operator algebras, invariants of knots and 3-manifolds, and number theory. 

The philosophy of representation theory tells us that to understand rings, we should study their modules. Lifting this to the level of categories motivates the study of 2-representations, or modules over tensor categories. In this learning seminar, we will study some classical and beautiful examples of such module categories. We will start with the basics of tensor categories, following [EGNO15]. We will try to make this brief. Then we will proceed with a tour of examples, with a particular focus on Verlinde categories and Soergel bimodules. In the second half of the course, we will discuss the basics of module categories, then examine the module categories for our running examples. Additional topics will be driven by participant interest/willingness to give talks. 

Meeting Information

The seminar will meet on Wednesdays from 10am-12pm during USyd's Semester 1. 

Talks will take place on the University of Sydney campus in Carslaw 830. 

Participants are invited to attend the talks in person - no recordings or live streaming will be organised. 

Notes will be posted on this website following each talk. 

Contact the organisers Anna Romanov and Alex Sherman for questions. 

Problem Sessions

Problem sessions to supplement the course will be held on Wednesdays from 2-4pm in the Quadrangle Seminar Room S441. 

Problem sessions will be led by Tasman Fell. 

All students and participants who are unfamiliar with the material of the course are highly encouraged to attend the problem sessions. 

Problem sets will be posted here. 

• Exercises Lecture 1 (Anna)

• Exercises Lecture 2 (Anna)

• Exercises Lecture 3 (Victor) 

• Exercises Lecture 4 (Alex)

• Exercises Lecture 5 (Joe) 

• Exercises Lecture 6 (Thomas) 

• Exercises Lecture 7 (Alex + Thomas)

• Exercises Lecture 8 (Tasman) 

• Exercises Lecture 9 (Joe) 

• Exercises Lecture 10 (Daniel) 

• Exercises Lecture 11 (Alex)

• Exercises Lecture 12 (Finn)

Schedule and Notes

Part 1: Tensor categories 

• • 21 February: Anna Romanov, Basics of tensor categories (notes)

  • 28 February: Anna Romanov, Basics of tensor categories part 2 (notes) 

  • 6 March: Victor Zhang, Diagrammatics of monoidal categories (notes)

  • 13 March:  Alex Sherman, Tannakian formalism (notes) 

  • 20 March: Joe Newton, Tannakian formalism (notes) 

  • 27 March: Thomas Dunmore, Running example 1: Verlinde categories (notes, slides) 

  • 3 April: NO SEMINAR  - USyd semester break

  • 10 April: Alex Sherman, Thomas Dunmore, Modular tensor categories (Alex's notes, Thomas's notes)

  • 17 April: Tasman Fell, Running example 2: Soergel bimodules (notes) 

  • 24 April: Joe Baine, The Dihedral cathedral (notes) 

Part 2: Module categories

• • 1 May: Daniel Dunmore, Basics of module categories (notes, slides) 

  • 8 May: Alex Sherman, Modules over Verlinde categories (notes) 

  • 15 May: NO SEMINAR - UNSW term break

  • 22 May: Finn Klein, Representation theory of symmetric groups (notes)

  • 29 May: 9:30-11:30am (*note special time) Bregje Pauwels, Chuang-Rouquier SL2 categorification (notes)

The End.

Resources

[EGNO15] Etingof, Gelaki, Nikshych, Ostrik: Tensor Categories 

[EMTW20] Elias, Makisumi, Theil, Williamson: Introduction to Soergel bimodules