# Categorification at roots of unity for pedestrians

The goal of this seminar is to understand the Khovanov-Qi, Elias-Qi approach how to categorify (small) quantum groups at (prime) roots of unity. Then, we try to understand what needs to be done to categorify the divided power version (note that in “the categorified root of unity case” one, in contrast to “the generic categorification case”, has to be very careful when it comes to Karoubi envelopes). To this end, we will have a regular meeting every Monday with one main speaker giving a 90 minutes talk about a particular part of their approach. Additionally, there will be a discussion/answering of questions session every second Friday.

### Contact

Catharina Stroppel email
Daniel Tubbenhauer email

### Where and when?

• Usual session:
• Seminarraum, Max-Planck-Institut für Mathematik (MPIM), Vivatsgasse 7, 53111 Bonn, Germany
• Every Monday from 16:15-17:45
• First meeting: Monday 18.04.2016
• Exceptions:
• The second talk will take place Friday 22.04.2016 from 13:30-15:00 in Seminar room 1.008, Mathematik-Zentrum
• There will be no meeting on Monday 16.05.2016, neither on Monday 27.06.2016
• There will be no question session on Friday the 03.06.2016
• Question session:
• Seminar room 1.008, Mathematik-Zentrum, Endenicher Allee 60, 53115 Bonn, Germany
• Every second Friday from 14:15-15:45
• First meeting: Friday 06.05.2016
• Preliminary meeting:
• Seminarraum, Max-Planck-Institut für Mathematik (MPIM), Vivatsgasse 7, 53111 Bonn, Germany
• Monday 04.04.2016, 16:15-17:45

### Literature

• [1] B. Elias and Y. Qi, “An approach to categorification of some small quantum groups II”, Adv. Math. 288 (2016), 81-151, ArXiv link: http://arxiv.org/abs/1302.5478
• [2] B. Elias and Y. Qi, “A categorification of quantum $\mathfrak{sl}(2)$ at prime roots of unity”, ArXiv link: http://arxiv.org/abs/1503.05114
• [3] A. Ellis and Y. Qi, “The differential graded odd nilHecke algebra”, ArXiv link: http://arxiv.org/abs/1504.01712
• [4] M. Khovanov, “Hopfological algebra and categorification at a root of unity: the first steps”, ArXiv link: http://arxiv.org/abs/math/0509083
• [5] M. Khovanov, A.D. Lauda, M. Mackaay and M. Stošić, “Extended graphical calculus for categorified quantum $\mathfrak{sl}(2)$”, Mem. Amer. Math. Soc. 219 (2012), no. 1029, vi+87 pp, ArXiv link: http://arxiv.org/abs/1006.2866
• [6] M. Khovanov and Y. Qi, “An approach to categorification of some small quantum groups”, Quantum Topol. 6 (2015), no. 2, 185-311, ArXiv link: http://arxiv.org/abs/1208.0616
• [7] A.D. Lauda, “An introduction to diagrammatic algebra and categorified quantum $\mathfrak{sl}(2)$”, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 2, 165-270, ArXiv link: http://arxiv.org/abs/1106.2128
• [8] G. Lusztig, “Introduction to quantum groups”, Reprint of the 1994 edition. Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010
• [9] Y. Qi, “Hopfological Algebra”, Compos. Math. 150 (2014), no. 1, 1-45, ArXiv link: http://arxiv.org/abs/1205.1814

### Schedule

• Monday 18.04.2016, Speaker: Daniel Tubbenhauer, Title: Categorification at a root of unity in a nutshell
• Friday 22.04.2016, Speaker: Arik Wilbert, Title: Quantum groups at roots of unity - the generic quantum group, the small quantum group and the divided power quantum group
• Monday 25.04.2016, Speaker: Tim Seynnaeve, Title: “Generic” categorification: the KL-R 2-category I
• Monday 02.05.2016, Speaker: Deniz Kus, Title: “Generic” categorification: the KL-R 2-category II
• Monday 09.05.2016, Speaker: Sondre Kvamme, Title: Basics of $p$DG homological algebra I
• Monday 23.05.2016, Speaker: Olaf Schnürer, Title: Basics of $p$DG homological algebra II
• Monday 30.05.2016, Speaker: Jacinta Torres, Title: $p$-differentials on the KL-R 2-category
• Monday 06.06.2016, Speaker: Ruslan Maksimau, Title: “Fantastic filtrations”
• Monday 13.06.2016, Speaker: Catharina Stroppel, Title: Decategorification: the small quantum group
• Monday 20.06.2016, Speaker: Tina Kanstrup, Title: The first step towards the divided power quantum group: Grothendieck groups and the Karoubi envelope in the $p$DG case
• Monday 04.07.2016, Speaker: Leonardo Patimo, Title: A brief review of thick calculus
• Monday 11.07.2016, Speaker: Daniel Tubbenhauer/Lars Thorge Jensen, Title: $p$DG thick calculus
• Monday 18.07.2016, Speaker: Michael Ehrig, Title: Decategorification: the divided power quantum group

### Abstracts

Daniel Tubbenhauer (11.04.2016),
Title: Categorification at a root of unity in a nutshell,
Abstract: The main point of this talk is to explain the motivation and goal of this seminar. We will explain why categorification at a root of unity is interesting by giving a short history of the subject. Then we will explain the main ideas behind the “How to”:

• How can one categorify the quantum group in the generic case using the KL-R approach
• We explain the difference between the generic quantum group and its root of unity cousins
• We explain Khovanov's main idea what can be done if some powers of generators are zero
• The main tool for this is the so-called $p$DG-structure
This talk is an overview - by definition it is a summary and does not contain details or technicalities.
Sources: [1], [2], [3], [4], [6], [9]
Some funny pictures: click

Arik Wilbert (18.04.2016),
Title: Quantum groups at roots of unity - the generic quantum group, the small quantum group and the divided power quantum group,
Abstract: The goal of this talk is to recall the basic definitions related to quantum groups and their “dot-versions”. Then two different specializations at roots of unity are explained: the so-called small quantum group and the divided power quantum group (and their “dot-versions”). The focus lies on the $$\mathfrak{sl}_2$$ case.
Sources: [1, Subsection 6.1], [2, Subsection 3.5], [7, Subsection 1.2], [8, Chapter 36]

Tim Seynnaeve (25.04.2016),
Title: “Generic” categorification: the KL-R 2-category I,
Abstract: This talk recalls the categorification of (generic) quantum $$\mathfrak{sl}_2$$ by introducing the diagrammatic calculus due to Khovanov-Lauda.
Sources: [7, Sections 2 and 3]

Deniz Kus (02.05.2016),
Title: “Generic” categorification: the KL-R 2-category II,
Abstract: This talk, which continues to present the case of the generic quantum $$\mathfrak{sl}_2$$ categorification, presents the necessary steps to show that the Grothendieck group of the KL-R 2-category gives the desired result.
Sources: [7, Sections 2 and 3]

Sondre Kvamme (09.05.2016),
Title: Basics of $p$DG homological algebra I,
Abstract: The main technical tool for categorification at roots of unity is the so-called $p$DG homological algebra. The purpose of this talk is to give the basic definitions regarding this $p$DG homological algebra.
Sources: [1, Section 2], [2, Subsections 4.5, 4.6, 4.7 and 4.8]

Olaf Schnürer (23.05.2016),
Title: Basics of $p$DG homological algebra II,
Abstract: In this talk more $p$DG homological algebra (needed later on) is introduced.
Sources: [1, Section 2], [2, Subsections 4.5, 4.6, 4.7 and 4.8]

Jacinta Torres (30.05.2016),
Title: $p$-differentials on the KL-R 2-category,
Abstract: This talk introduce the $p$DG structure on the KL-R 2-category in the $$\mathfrak{sl}_2$$ case.
Sources: [1, Section 4]

Ruslan Maksimau (06.06.2016),
Title: “Fantastic filtrations”,
Abstract: The main tool to understand the Grothendieck group in the root of unity case are certain filtrations which are discussed in details in this talk.
Sources: [1, Section 5]

Catharina Stroppel (13.06.2016),
Title: Decategorification: the small quantum group,
Abstract: The purpose of this talk is to make precise how to categorify the small quantum group at roots of unity by giving the details about how to compute the Grothendieck group.
Sources: [1, Section 6]

Tina Kanstrup (20.06.2016),
Title: The first step towards the divided power quantum group: Grothendieck groups and the Karoubi envelope in the $p$DG case,
Abstract: In abelian categorification it is a typical practice to take the Karoubi envelope), while the $p$DG world requires a great deal more caution (as we will see later). In this talk the basics about Karoubi envelopes are discussed - and then the changes which might appear in the $p$DG case.
Sources: [2, Section 4], [5, Section 3]

Leonardo Patimo (04.07.2016),
Title: A brief review of thick calculus,
Abstract: The so-called “thick calculus” is a method to diagrammatically present the Karoubi envelope in case of categorified quantum $$\mathfrak{sl}_2$$. This talk introduces the necessary diagrammatics to understand the basics of this calculus.
Sources: [5, Section 4]

Daniel Tubbenhauer/Lars Thorge Jensen (11.07.2016),
Title: $p$DG thick calculus,
Abstract: This talk introduces a “$p$DG thick calculus” which enriches the “thick calculus” from the previous talk.
Sources: [2, Section 5]

Michael Ehrig (18.07.2016),
Title: Decategorification: the divided power quantum group,
Abstract: This talk makes it precise how one can see that the decategorification gives back the divided power version (instead of the small quantum group).
Sources: [2, Section 6]