# GEARS Seminar

The Glasgow Edinburgh Algebra Research Student seminar is aimed at postgraduates at Glasgow and Edinburgh Universities who use algebra in their work. We meet roughly three times per semester and speakers present an important paper in their area of research and explain how this relates to their own work.

The seminar is currently organised by Simon Crawford and Noah White. Get in touch if you have any questions.

## Upcoming meetings

### Glasgow, 30 March 2016

 When: 4pm Wednesday 30 March 2016 Where: Lecture theatre 507, Boyd Orr building, University of Glasgow. Speakers: Angela Tabiri (Glasgow), and Gwendolyn Barnes (Heriot-Watt) (details of the talks are below)

Please note the unusual location. The Boyd Orr Building is next to the Mathematics department

Angela Tabiri Quantum homogeneous spaces

This talk will be based on the papers:

Abstract/reading instructions: Quantum homogeneous spaces are Hopf algebras with additional properties. In this seminar, I will define and give an example of a quantum homogeneous space. Then I will end with some examples of planar curves which are quantum homogeneous spaces.

Gwendolyn Barnes The Quantum Group Revolution

This talk will be based on the paper:

Abstract/reading instructions: The theory of connections on fiber bundles underlies our modern understanding of classical field theories such as gravity and Yang-Mills theory. Recent observations suggest that the corresponding quantum theories are noncommutative and nonassociative. An abstract notion of geometry is required in order to describe these quantum field theories in a manner that is compatible with the classical descriptions. In work together with Alexander Schenkel and Richard J. Szabo we have appealed to the tools of category theory and topos theory to better understand the mathematical structures underlying noncommutative and nonassociative deformations of spacetime geometry [arXiv:1601.07353, arXiv:1507.02792, arXiv:1409.6331].

## Previous meetings

### Edinburgh, 15 February 2016

 When: 5:15pm Monday 15 February 2016 Where: Room LG.11, David Hume Tower , University of Edinburgh. Speakers: Getachew Alemu Demessie (Heriot-Watt), and Tim Weelinck (Edinburgh) (details of the talks are below)

Please note the unusual location. The Boyd Orr Building is next to the Mathematics department

Getachew Alemu Demessie Smooth 2-groups (The string 2-group model by Schommer-Pries)

This talk will be based on the papers:

Abstract/reading instructions: In this talk I will discuss categorical groups by using the concept of internalization. In particular, I will explain the construction of the string 2-group model by Schommer-Pries, which is a 2-group object in a bicategory. Finally, I will mention how this string 2-group can be Lie differentiated to the known Lie 2-algebra.

Tim Weelinck The Quantum Group Revolution

This talk will be based on the paper:

Abstract/reading instructions: We tell the story of Drinfeld’s 1986 ICM address, undoubtedly one of the milestones in mathematics with a whopping 676 references on MathSciNet. In his address Drinfeld introduces objects he calls Quantum Groups’, certain deformations of the universal enveloping algebra of a Lie algebra, that carry canonical solutions to the Yang-Baxter equation.

We will begin by telling the history of the mathematical (and physical) theories that Drinfeld intertwined to create his quantum groups. In passing we will discuss the relevance of the Yang-Baxter equation in physics (and math). Thereafter, we will take our time to discuss quantum groups, as introduced by Drinfeld in his paper e.g. introducing Lie bialgebras, Poisson-Lie groups, quantizations of these, etc.

Nowadays quantum groups are connected to a wide variety of mathematical topics: conformal field theory, modular representations of Lie algebras, algebraic number theory and even low-dimensional topology. If time permits we will touch on these connections. Alternatively we can have a light-hearted discussion on why these objects are called quantum groups’.

### Glasgow, 15 January 2016

 When: 3:45pm Friday 15 January 2016 Where: Room 513, Boyd Orr Building , University of Glasgow. Speakers: Tomasz Przezdziecki and Simon Crawford (details of the talks are below)

Please note the unusual location. The Boyd Orr Building is next to the Mathematics department

Tomasz Przezdziecki KLR algebras and geometric representation theory

This talk will be based on the paper:

Abstract/reading instructions: The main goal of the talk is to explain the geometric construction of KLR (quiver Hecke) algebras due to Vasserot & Varagnolo. These algebras were introduced by Khovanov & Lauda and Rouquier by combinatorial and algebraic means. Research on KLR algebras has been motivated by their role in the categorification of quantum groups and their connection to affine and cyclotomic Hecke algebras. We will briefly review classical geometric representation theory, recalling the Steinberg variety and the construction of the group algebra of a Weyl group in its homology. We will then sketch how to modify this construction to obtain a KLR algebra. The interesting new features include the use of quiver flag varieties and equivariant homology.

Simon Crawford Deformations of Kleinian singularities and singularity categories

This talk will be based on the papers:

Abstract/reading instructions: In 1998, Crawley-Boevey and Holland introduced a family of deformations of the coordinate rings of Kleinian singularities, which may be viewed as a generalisation of earlier work by Hodges. They also introduced the deformed preprojective algebra, and proved that their existed Morita equivalences and, under suitable hypotheses, isomorphisms between these two families of deformed algebras. This allowed them to deduce a number of nice ring theoretic and homological properties. I will give an overview of their results, with an emphasis on examples, and will also try to explain how their results have motivated my research.

### Edinburgh, 5 November 2015

 When: 3:45pm Thursday 5 November 2015 Where: ICMS, Edinburgh. Speakers: Bernard Bainson and Salvatore Dolce (details of the talks are below)

Bernard Bainson Ordered Groupoids and Left Cancellative Categories

This talk will be based on the papers:

Abstract/reading instructions: Leech (1987) in his work presented the idea of constructing an inverse semigroup from small categories. Loganathan describes the cohomology of inverse semigroups via an associated category which turns out to have some advantages over the traditional approach in homological theory. In this presentation, we shall consider the correspondence between ordered groupoids and left cancellative categories due to Lawson and state some applications.

Salvatore Dolce Vinberg Monoid and Reductive Semigroups

This talk will be based on the papers:

Abstract/reading instructions: The Vinberg Monoid has its origin in the theory of algebraic semigroups and it represents a minimal element in a special family of monoids. I will introduce this object as Vinberg did in his original paper “On reductive semigroups” in which he worked in the setting of complex algebraic groups. Then I will discuss how to generalise the theory over any algebraically closed field using a more geometric approach via spherical varieties. Finally, time permitting, I will tell you a nice interpretation of the Vinberg Monoid as a certain moduli space of principal bundles which could have an analogous in an infinite dimensional setting.

### Edinburgh, 14-18 September 2015

 When: 3pm 15 September 2015 Where: ICMS, Edinburgh. Speaker: Claire Amiot (details of the talks are below)

Claire Amiot (Institut Fourier, Grenoble) Cluster categories for algebras of global dimension 2 and cluster-tilting theory

This will be a short series of lectures on generalised cluster categories. This will be followed later in the week by a research seminar.

Abstract: In this talk I will present basic results of cluster-tilting theory developed in [Iyama Yoshino 2008: Mutation in triangulated categories and Rigid CM modules] and [Buan-Iyama-Reiten-Scott 2009: Cluster structures for 2-Calabi-Yau categories]. I will explain how these results were a motivation for generalising the construction of cluster categories. I will first recall the motivation and definition of the acyclic cluster category due to Buan Marsh Reineke Reiten Todorov in 2006, and then focus to the construction of the generalised cluster category associated with algebras of global dimension 2 [Amiot 09]. Then I will explain how cluster-tilting theory can apply in classical tilting theory via graded mutation in a joint work with Oppermann.

### Glasgow, 16 April 2015

 When: 4pm Thursday 16 April 2015 Where: School of Mathematics, University of Glasgow. Speakers: Astrid Jahn and Cesar Lecoutre (details of the talks are below)

Astrid Jahn The finite dual of crossed products

This talk will be based on Astrid’s thesis and the papers:

Abstract/reading instructions: In finite dimensions, Hopf algebras have very nice duality properties: the vector space dual of any finite-dimensional Hopf algebra is also a Hopf algebra, and in this setting the canonical isomorphism between a vector space V and the dual V** is in fact an isomorphism of Hopf algebras. In infinite dimensions, this breaks down: the vector space dual of a Hopf algebra is no longer itself a Hopf algebra in general. Although there does exist a subspace of the vector space dual called the “finite dual” which is always a Hopf algebra and can be used as a replacement, this does not satisfy many of the properties known in the finite-dimensional case. Most obviously, finite duals need not preserve the size of a Hopf algebra: on the one hand there exist infinite-dimensional Hopf algebras whose finite duals are one-dimensional, on the other countably infinite-dimensional Hopf algebras whose finite dual is uncountable.

My aim in this talk is to better understand the finite dual by understanding how the structure of the finite duals of Hopf algebras which are “built up” out of smaller (Hopf) algebras relates to the finite duals of their components. I look at two different types of such Hopf algebras: first, Hopf algebras which decompose as crossed products (a generalised Hopf algebra version of the semidirect product), second, Hopf algebras which are finite modules over some central Hopf subalgebra.

Reading list: It would be helpful if people had a good understanding of the definition of a Hopf algebra before the talk - this can for instance be found in the book “Hopf algebras and their actions on rings” by Susan Montgomery - however, I will most likely be giving a brief overview of Hopf algebras at the start. Important definitions to know in order to follow the examples are those of group algebras, universal enveloping algebras of Lie algebras and coordinate rings of algebraic varieties, and it would be useful if people had at least a vague idea of the quantised version (quantum group) of the last two.

Other relevant papers include Donkin’s “On the Hopf algebra dual of an enveloping algebra” for a result regarding the finite dual of crossed products which I generalise, Blattner, Cohen and Montgomery’s “Crossed products and inner actions of Hopf algebras” for the definition of a crossed product (this can also be found in Montgomery’s book “Hopf algebras and their actions on rings”) and Brown and Zhang’s “Prime regular Hopf algebras of GK-dimension one” for the classification of prime affine Hopf algebras of Gelfand-Kirillov dimension one, which I will be making reference to in the talk.

Cesar Lecoutre About the Gel’fand-Kirillov Conjecture

This talk will be based on the paper:

Abstract/reading instructions: Classifying the objects of a given nature is one of the goals of algebra. When classifying rings one often tries first to classify them up to birational equivalence, i.e. to classify their (skew)-field of fractions. We present the 1966 paper of Gel’fand and Kirillov, where the authors study this question for enveloping algebras of Lie algebras. They conjectured that an enveloping skewfield (the skewfield of fractions of an enveloping algebra of a (certain) Lie algebra) is isomorphic to a Weyl skewfield, a rather simple skewfield. Moreover they solve the isomorphism problem for Weyl skewfields. We will also survey the development around their conjecture as well as we will present its quantum and Poisson versions.

Suggested topics for reading: -non commutative localisation -enveloping algebra of Lie algebra -good example: think of U(sl2) and invert everything. Can you simplify the relations? Can you find generators x,y,z such that xy-yx=1 and z is central?

The first two topics will be briefly covered, they can be found in Noncommutative Noetherian Rings by McConnel and Robson.

### Edinburgh, 29 Jan 2015

 When: 4pm Thursday 29 January 2015 Where: ICMS, University of Edinburgh. Speakers: Przemek Pobrotyn and Paul Slevin (details of the talks are below)

Przemek Pobrotyn The Golod-Shafarevich theorem

This talk will be based on the papers:

Abstract/reading instructions: In their 1964 paper ‘On class field towers’ Shafarevich and his student Golod solved a long standing number theoretical problem, the class field tower problem. Their main tool was what is now referred to as the Golod-Shafarevich inequality. It can be formulated in many different categories and it relates certain growth function of an object in a category with certain data coming from the presentation of that object by generators and relators. Shortly afterwards Golod and Shafarevich used their newly created tool to solve Kurosh/Levitski Problem and subsequently gave the negative answer to the General Burnside Problem. In this talk I will explain the GS inequality and give an elementary proof of it, as presented in Regev’s ‘The Golod Shafarevich counter–example without Hilbert series’. I will also describe the Kurosh and Burnside problems and show how they can be settled to negative using the aforementioned inequality.

Paul Slevin Cyclic Homology and Duplicial Objects

This talk will be based on the paper:

Abstract/reading instructions: Cyclic homology was invented by Alain Connes and was intended to generalise the de Rham (co)homology of manifolds. In my talk, I will give the definition of the cyclic (and periodic cyclic) homology of a mixed complex and look at subclasses of examples that give rise to cyclic homology theories, usually with the help of a left adjoint functor. In particular, I will define duplicial objects - these are simplicial objects with a so-called “extra degeneracy” that is compatible with the face and degeneracy maps. Miraculously, if we are inside an abelian category, these objects correspond to duchain complexes - that is, things which are simultaneously chain and cochain complexes with no compatibility conditions between the differentials whatsoever. This correspondence is known as the “Dwyer-Kan” correspondence and is closely related to the “Dold-Kan” correspondence. I will finish by discussing some recent research, specifically that concerned with the construction of duplicial objects (and hence new cyclic homology theories).

1. Familiarise yourself with simplicial objects, e.g. page 4 here

2. See Wikipedia for the definition of the Hochschild simplicial object. For the choice $M = A$, define an extra degeneracy $s_{n+1}$ in each degree that inserts 1 in the first slot. Check that the duplicial relations (see 2.1 of Dwyer/Kan [also note the typo, $j-1$ should read $j-i$]) are satisfied when we add this extra degeneracy to the simplicial structure.

### Glasgow, Thursday 4 Dec December 2014

 When: 4pm Thursday 4 December 2014 Where: Room 515, School of Mathematics, University of Glasgow. Speakers: Haris Stylianakis and Noah White (details of the talks are below)

Haris Stylianakis Congruence subgroups of braid groups.

This talk will be based on the papers:

Abstract/reading instructions: We can consider the symmetric group as a quotient of the braid group over the normal closure of a braid generator squared. Consider the normal closure of a braid generator to the power of an arbitrary integer. Then, is the quotient finite? Coxeter in his paper “factor groups of braid groups” proved that the answer is not always positive. Are there finite index normal subgroups of braid groups, that include a normal closure of a braid generator to an arbitrary power? If so, what is the quotient group? Wajnryb in his paper “A braidlike presentation of $Sp(n, p)$” provided a presentation of the symplectic group over a finite field in terms of the braid group. This presentation allow us to study the normal subgroups of braid groups on which the quotient is the symplectic group. In this talk we will examine these normal subgroups, and we will provide a geometric interpretation in terms of elements of mapping class groups of surfaces.

It would be beneficial to read the definitions of the symplectic group and to know a nice presentation for braid groups. I would also suggest to look at the definition of “mapping class group”. You can find these definitions in Benson Farb and Dan Margalit’s book:

• Chapter 2: definition of mapping class group
• Chapter 6: definition of symplectic group
• Chapter 9: definition of braid group

Noah White The representation theory of the symmetric groups via the action of a “maximal torus”.

This talk will be based on the paper:

Abstract/reading instructions: The representation theory of the Symmetric groups is a very old subject going back to work of Frobenius and Schur in the 19th century and then, importantly, taken up by Young at the turn of the century. Whilst the irreducible representations can be constructed, very little is known about them, even today. The classical approach is to reason that since a finite group must have an irreducible representation for every (p-regular) conjugacy class we should look for irreducible representations labelled by partitions. This approach, while effective, is somewhat artificial in the sense that it forces the combinatorics of partitions and tableaux upon us.

I will outline Okounkov and Vershik’s modern approach which gives the subject a Lie theoretic flavour and demonstrates the origin of the combinatorics and exactly how this controls the representation theory. Time permitting I will explain how this approach leads to a much better understanding of the representation theory (mod p) through categorification and how this has been applied to deduce results not known previously for the symmetric groups and many related algebras.

See pages 2 and 3 of these notes for the definition of the Specht modules. Try carrying out the construction for the partitions $(2), (1,1)$ (one dimensional representations of $S_2$) and $(2,1)$ (a two dimensional representation of $S_3$). Now decompose the $S_3$ rep $(2,1)$ as a direct sum of $S_2$ reps.

### Edinburgh, Thursday 13 November 2014

 When: 4pm Thursday 13 November 2014 Where: 11.18 (level 11) David Hume Tower, University of Edinburgh Speakers: Ana Rovi and Joe Karmazyn (details of the talks are below)

Ana Rovi Differential forms on general commutative algebras.

This talk will be based on the paper:

Abstract/reading instructions: We will use some results in that paper to develop examples of Hopf algebra structures on the enveloping algebra of a certain generalisation of Lie algebras. We show that this construction shows interesting differences with respect to the classical case.

Joe Karmazyn Constructing moduli spaces of quiver representations.

This talk will be based on the paper:

Abstract/reading instructions: Given a finite dimensional path algebra of a quiver with relations this paper constructs a variety that is the moduli space of certain representations of this quiver with relations. This associates various geometric spaces to these noncommutative algebras. The construction itself is very concrete, and I will try and give lots of examples.

Constructing this space as a (fine) moduli space is more powerful than just constructing it as a variety as it also produces an associated family of vector bundles. Time permitting, I will try and mention how this additional structure has been used to establish deeper links between these noncommutative algebras and geometry.

While I will recall all the required definitions it might be helpful to try and familiarise yourself with quivers in general and particularly with finite dimensional quiver representations. Finding all representations of the Kronecker quiver with dimension vector (1,1) up to isomorphism would be a helpful exercise.

### Glasgow, Thursday 23 October 2014

 When: 4pm Thursday 23 October 2014 Where: Room 516, School of Mathematics, University of Glasgow. Speakers: Chris Campbell and Paul Gilmartin (details of the talks are below)

Chris Campbell:
The Gerstenhaber Algebra Structure on Hochschild Cohomology for Koszul Algebras.

This talk will be based on the papers:

Abstract/reading instructions: The first paper establishes a Gerstenhaber algebra structure on Hochschild cohomology and I will explain this, people may want to wikipedia this just to know what to expect. The paper is mostly technical lemmas establishing the necessary properties and can be fairly lightly skim read with that in mind. As for the second paper, I will concentrate on the first section, as this is the motivation for my research. I will mostly be discussing the definitions and giving them some slightly more modern context. This boils down to what a deformation is for an associative algebra, and how Hochschild cohomology is related to it. I won’t assume anyone has read anything in these but it may make for a softer landing.

Paul Gilmartin:
Connected Hopf algebras of finite GK dimension.

This talk will be based on the papers:

Abstract/reading instructions: My talk will begin probably with the definition of a Hopf algebra (but it would probably be better if people knew this beforehand). I shall then proceed to (briefly) explain the meaning of “connected” and “finite GK dimension”, why they can be seen to be natural conditions to impose on Hopf algebras (from both a geometric and algebraic point of view) and then go on to look at a couple of explicit and motivating examples of Hopf algebras satisfying these conditions.

With the background material covered, my next goal will be to discuss some of the results (without much mention of their proofs) proved by Wang, Zhang and Zhuang in the first two of the aforementioned papers, which establish many nice ring-theoretic facts about these Hopf algebras. As time permits, I shall then talk about my own research, which has been to generalise the results of Wang, Zhang and Zhuang to “coideal subalgebras” of connected Hopf algebras.

I do not expect anyone to read any of the papers I’ve listed. What would be good is if people knew the definitions of a filtered ring and the associated graded ring of a filtered ring, but I’ll probably explain these things anyway. For certain parts of the talk it may also be useful to be vaguely familiar with Hilbert’s Nullstellensatz, although this probably won’t be essential.

## Funding

The GEARS Seminar is funded by the Institute fo Academic Development at the University of Edinburgh. If you are a participant and you want to enquire about the reimbursement procedure, please email Noah White.