Lie Superalgebras and categorification (Spring 2023)

Organized by Cailan Li and Alvaro Martinez. Please email me at to join the mailing list.

This is a learning seminar on superalgebras and their categorifications. The plan is to first give an overview of superalgebras and their representation theory, and then cover recent developments in the categorification program of super-stuff and related topics, as you can see in the outline below.

Outline of the seminar.

Talks will roughly be 45 minutes followed by a 5 minute break followed by 45 more minutes. Please email me at if you'd like to give a talk.


Wed Jan 25 Cailan Li
Lie superalgebras: Fundamentals

We introduce Lie superalgebras and explain the subtle differences with usual Lie algebras that occur in the theory. In particular, the presence of isotropic odd roots will force us to decorate the Dynkin diagrams associated with usual Lie algebras in order to adequately describe the structure of Lie superalgebras.

Notes. Live notes.
Tue Jan 31 at 4.30pm Fan Zhou
Lie superalgebras: Representations

We discuss the universal enveloping algebra and weight theory for (basic) Lie superalgebras. We will classify all finite-dimensional simples of gl(m|n) and also begin discussing central characters in anticipation of linkage next week.

Notes. Live notes.
Week of Feb 6: No meeting
Wed Feb 15 at 5.40pm Alvaro Martinez
Categorification of quantum \(\mathfrak{gl}(1|2)^+\)

We go over Khovanov's categorification of the positive half of the Lie superalgebra \(U_q^+(\mathfrak{gl}(1|2))\).

Notes. Live notes.
Wed Feb 22 at 5.40pm Alvaro Martinez
Categorification of quantum \(\mathfrak{gl}(1|2)^+\) (continued)

We will clarify the notion of taking the Grothendieck group of a dg ring. Then we will finish proving that the construction from last time categorifies \(U_q(\mathfrak{gl}(1|2)^+)\), by exploring its similarity with the KLR algebras setting.

Notes. Live notes.
Wed Mar 1 at 5.40pm Cailan Li
Braid Group Actions and PBW Bases for \(U_q(\mathfrak{gl}(m|1))\)

We first review the construction of Braid Group Actions and PBW Bases for U_q(\mathfrak{g}) in the classical setting. We then do the same but for \(U_q(\mathfrak{gl}(m|1))\) with an eye towards the construction of the crystal basis for \(U_q(\mathfrak{gl}(m|1)^+)\).

Notes. Live notes.
Wed Mar 22 at 5.40pm Cailan Li
Canonical Bases for U_q(gl(m|1))

We finish up the construction of PBW bases for U_q(gl(m|1)) and then review Lusztig's construction of the canonical basis for U_q(g) in type ADE via PBW bases. We then move to U_q(gl(m|1)) and finish with some examples.

Notes. Live notes.
Wed Mar 29 at 5.40pm Alvaro Martinez
Categorification of tensor powers of the vector representation for gl(1|1)

Abstract: We will go over Sartori's paper on categorification of V^{\otimes n} for gl(1|1). Time permitting, we will define the Sartori algebras whose module categories achieve the same categorification.

Notes. Live notes.
Wed Apr 12 at 5.40pm Andrew Manion
Ozsvath-Szabo bordered algebras and subquotients of category O (joint with A. Lauda)

Abstract: I will summarize and discuss joint work with Aaron Lauda relating Sartori's and Ozsvath-Szabo's categorifications of tensor powers of the vector representation of U_q(gl(1|1)). In particular, I will discuss the motivations behind Ozsvath-Szabo's theory and how they relate to Sartori's motivations, then sketch a bit about Ozsvath-Szabo's theory using Sartori's theory as a reference point. I will comment on the main result of my work with Lauda, that Sartori's categorification algebra is a quotient of Ozsvath-Szabo's by a naturally defined ideal (compatibly with categorified actions of U_q(gl(1|1))), and mention avenues for future investigation if time permits.

Notes. Live notes.
Wed Apr 19 at 5.40pm Fan Zhou

Abstract: TBD

Notes. Live notes.
Wed Apr 26 at 5.40pm Alvaro Martinez
KLR algebras for non-simply laced types

Abstract: We will see in detail how KLR algebras categorify the quantum Serre relations, with an eye towards the categorification of Lie superalgebras.

Notes. Live notes.
Wed May 03 at 5.40pm Cailan Li
Categorification of U(\widehat{sl_p})

We first prove preliminary results about the socle and head of certain modules over the dAHA /k to build up to the proof that the character map is injective. Time permitting we will then show how to categorify the Type A Serre relations and how to categorify U(\widehat{sl_p} ) using modules over U(\widehat{sl_p} ) where p is the characteristic of the field k.

Notes. Live notes.