Below you will find titles, abstracts and electronic versions of talks and posters I have given or am due to give, most recent first. Electronic versions are PDF files.

2019 | 2018 | 2017 | 2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2010 | 2009 | 2008 | 2007 | 2006 | 2005 | 2004 | 2003 | 2002

8th November 2019:
*Leveraging cluster structures*

Algebra, Geometry and Topology seminar, School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury.

*For an abstract, please see my talk at Quantum Flag Manifolds in Prague below.*

16th-20th September 2019:
*Leveraging cluster structures*
*Quantum Flag Manifolds,* Charles University, Prague, Czech Republic.

*Abstract:* We now know that many important families of algebras admit quantum cluster algebra structures, including many related to quantum flag manifolds. So, what can we learn about these algebras from their cluster structures?

There are theorems telling us that certain behaviours in cluster structures imply ring-theoretic properties: perhaps the best known of these is that acyclic cluster algebras are Noetherian. What new information can we get from the cluster structure?

If our cluster algebra is of finite type, we know essentially everything about the cluster structure. Infinite types are more challenging, and most geometrically-derived examples are infinite type. We need ways of looking at cluster algebras that can see global properties: two such ways are gradings and categorification. We will discuss some results in these directions and pose some questions for further study.

*Slides:* prague19.pdf (layers)

23rd February 2019:
*A smörgåsbord of gradings in cluster theory*

Algebra and Geometry Seminar, 2.00pm, Department of Mathematics and Statisitcs, Lancaster University.

9th October 2018:
*A smörgåsbord of gradings in cluster theory*

Oberseminare Algebra, Darstellungstheorie und Homologische Methoden, 2.00pm, Institut für Algebra und Zahlentheorie, Universität Stuttgart, Germany.

27th March 2018:
*Recovering automorphisms of quantum spaces*

MAXIMALS seminar, School of Mathematics, University of Edinburgh.

*For an abstract, please see my talk at the London Algebra Colloquium below.*

14th February 2018:
*Recovering automorphisms of quantum spaces*

Algebra seminar, 4.00pm, School of Mathematics and Statistics, University of Glasgow.

*For an abstract, please see my talk at the London Algebra Colloquium below.*

1st February 2017:
*Recovering automorphisms of quantum spaces*

Algebra and Geometry seminar, 2.30pm, School of Mathematics, University of Bristol.

*For an abstract, please see my talk at the London Algebra Colloquium below.*

23rd January 2017:
*Recovering automorphisms of quantum spaces*

Algebra seminar, 4pm, Department of Mathematics, University of York.

*For an abstract, please see my talk at the London Algebra Colloquium below.*

1st November 2016:
*Recovering automorphisms of quantum spaces*

Algebra seminar, 2.15pm, Mathematical Institute, Oxford.

*For an abstract, please see my talk at the London Algebra Colloquium below.*

21st October 2016:
*Recovering automorphisms of quantum spaces*

Algebra seminar, School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury.

*For an abstract, please see my talk at the London Algebra Colloquium below.*

20th October 2016:
*Recovering automorphisms of quantum spaces*

London Algebra Colloquium, Department of Mathematics, City University London.

*Abstract:* It has long been expected, and is now proved in many important cases, that quantum algebras are more rigid than their classical limits. That is, they have much smaller automorphism groups. This begs the question of whether this broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative projective geometry, from which we see that the correct object to study is a groupoid, rather than a group, and maps in this groupoid are the replacement for automorphisms. I will illustrate this with the example of quantum projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

10th March 2016: *Recovering automorphisms of quantum spaces*

Pure Mathematics seminar, Department of Mathematics and Statistics, Lancaster University.

*For an abstract, please see my talk in Newcastle below.*

1st March 2016: *Recovering automorphisms of quantum spaces*

Algebra-Geometry Seminar, 3pm, School of Mathematics and Statistics, Newcastle University.

*Abstract:* It has long been expected and is now proved in many important cases that quantum algebras are more rigid than their classical limits. That is, they have much smaller automorphism groups. This begs the question of whether this broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative projective geometry, from which we see that the correct object to study is a groupoid, rather than a group. I will illustrate this with the example of quantum projective space.

This is joint work with Nicholas Cooney (Lancaster).

6th July 2015: *"What does that mean, precisely?" - learning to work with definitions*

Transition in year 1: Opportunities and challenges, Sharing Practice Day 2015, Lancaster University.

*Abstract:*The transition from school to university mathematics is notorious for being difficult. A significant part of this is because as well as more difficult material to comprehend, students need to adapt to a new way of thinking about what mathematics is and how they do it. In school, mathematics tends to be focussed on examples and methods, whereas university mathematics looks to address much broader questions - why is this true? - and, simultaneously, much more precise questions - what exactly do we mean by a particular notion.

It is therefore crucial that the first year curriculum is designed to help students make this double transition, in level and in philosophy. I will talk about one particular aspect, which my own first year module is tasked with addressing, namely the role of definitions in mathematics. If we want to assert that every whole number is either odd or even, first we have to fix what we mean by "whole number", "odd" and "even". We also have to agree these - if 3 is odd for me and even for you, we will struggle to have a common mathematical discourse.

So students have to come to appreciate that there are formal definitions, that these are fixed during a certain period (but may be refined at a later date) and that they have to know these definitions, in a very literal sense. This requirement for some declared knowledge, in order that a shared discourse can take place, is new to students making the transition to university study and is by no means restricted to mathematics. I will discuss how my curriculum design tries to help students appreciate its place in my subject in particular, and to help them adjust their learning approaches to suit.

10th-13th June 2015: *Recovering automorphisms of quantum spaces*

International Meeting of the American, European and Portuguese Mathematical Societies, Porto, Portugal.

*Abstract:* It has long been expected and is now proved in many important cases that quantum algebras are more rigid than their classical limits. That is, they have much smaller automorphism groups. This begs the question of whether this broken symmetry can be recovered.

I will outline an approach to this question involving groupoids and functors on them induced by 2-cocycle twists, focussing on the example of quantum projective space.

This is joint work with Nicholas Cooney (Lancaster).

16th March 2015:
*Clusters, combinatorics, grading, twisting and quantization*

"Algebraic Interfaces of Integrability", Workshop on Classical and Quantum Integrability, University of Leeds.

*Abstract:* Quantum cluster algebras are a natural noncommutative generalisation of cluster algebras. This noncommutativity is relatively mild: the main feature is that elements in the same cluster must commute up to a power of q, although elements from different clusters may have more complicated relationships. Many results for cluster algebras have direct parallels for the quantum case and we will discuss some of what is now known and not known for quantum cluster algebras.

We will also study gradings, show how they can be classified and use them to produce twisted cluster algebra structures. On the one hand, gradings bring out some beautiful combinatorics, in the form of frieze patterns, but we will also consider a more algebraic application of gradings, namely a "twisting" construction that is important for quantizing cluster algebra structures.

11th March 2015:
*Gradings on cluster algebras and associated combinatorics*

Pure Mathematics Colloquium, School of Mathematics and Statistics, University of Sheffield.

*Abstract:* When studying any class of rings or algebras, the existence of a grading often has a big impact on what can be said about the members of the class. In the few years since their inception, cluster algebras have been found in numerous places and have been shown to be responsible for a plethora of combinatorial patterns, but until very recently gradings on cluster algebras have not been considered in a systematic way.

In this talk, we will introduce gradings on cluster algebras and show how the intricate structure and combinatorics associated to cluster algebras allows us to find and classify gradings. We will look at cluster algebras of finite type and examine the gradings they admit, making use of cluster categories. Conversely, the gradings bring out some beautiful combinatorics of their own, in the form of tropical frieze patterns.

25th February 2015:
*Grading cluster algebras and categories *

67th Bristol-Leicester-Oxford Colloquium (BLOC), City University, London.

*Abstract:* The idea of a graded cluster algebra has been around since the origin of the subject but often implicitly. However in the study of quantum cluster algebras, gradings play a much more prominent role, which prompts us to examine the classical case more carefully.

We will transfer a definition of Gekhtman, Shapiro and Vainshtein into a more algebraic setting, to obtain the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients and give a full classification.

Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics - namely tropical frieze patterns - on the Auslander-Reiten quivers of the categories.

*Slides:* BLOC15.pdf

27th January 2015:
*Grading cluster algebras and categories*

Algebra seminar, 3pm, MALL, Physics Research Deck (Room 9.301), Department of Pure Mathematics, University of Leeds.

*Abstract:* The idea of a graded cluster algebra has been around since the origin of the subject but often implicitly. However when we examine the quantum version of cluster algebras, gradings play a central role, which prompts us to examine the classical case more carefully.

We will transfer a definition of Gekhtman, Shapiro and Vainshtein into a more algebraic setting, to obtain the notion of a multi-graded cluster algebra. We can then study gradings for finite type cluster algebras without coefficients and give a full classification.

Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics - namely tropical frieze patterns - on the Auslander-Reiten quivers of the categories.

*Slides:* leeds15.pdf

17th-18th October 2014:
*Quantizing cluster algebra constructions*

LMS Workshop on Cluster Algebras and Preprojective Algebras, School of Mathematics, Cardiff University.

*Abstract:* In order to obtain quantum analogues of various cluster algebras, we need quantum versions of some classical cluster algebra constructions. This can be done but the noncommutativity of the quantum setting means that the classical methods require careful reinterpretation and become more technically involved.

I will discuss some examples, concentrating mostly on the necessary ingredients for our work with Stephane Launois showing that quantum Grassmannians are quantum cluster algebras. In particular, that work builds on the categorification of cluster structures on (quantum) coordinate rings of big cells of partial flag varieties using preprojective algebra modules, due to Geiss, Leclerc and Schroer. We will see that this categorification includes essentially all of the information needed for the (quantum) cluster structure on Grassmannians, even though it does not quite categorify the latter in the usual sense.

21st-22nd July 2014:
*Cluster algebras and their quantum analogues* (4 lectures)

Kent Algebra Days PGR, School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury.

*Abstract:*
Cluster algebras are currently playing a prominent role in several areas of mathematics, including Lie theory, representation theory and geometry. We will introduce cluster algebras and explain some basic algebraic constructions for them, including morphisms, subalgebras and quotients. We will also study gradings, show how they can be classified and use them to produce twisted cluster algebra structures. We also explain some of the major theorems relating to cluster algebras, regarding their combinatorial and algebraic structure and in particular the main classification theorem of Fomin-Zelevinsky.

Quantum cluster algebras are a natural noncommutative generalisation of cluster algebras. This noncommutativity is relatively mild: the main feature is that elements in the same cluster must commute up to a power of q, although elements from different clusters may have more complicated relationships. Many results for cluster algebras have direct parallels for the quantum case so we will treat these alongside the classical case.

We will conclude with looking at two important classes of quantum cluster algebras in detail: quantum matrices, following the representation-theoretic approach of Geiss-Leclerc-Schroer, and quantum Grassmannians.

*Lecture notes:* kentPGR14.pdf

2nd June 2014:
*Clusters, combinatorics, grading, twisting and quantization*

Oberseminar zur Algebra und Algebraischen Kombinatorik, 4pm, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Germany.

*Abstract:*
Cluster algebras are currently playing a prominent role in several areas of mathematics, including Lie theory, representation theory and geometry (via mirror symmetry). We will introduce gradings on cluster algebras in a very natural way and show how to find and classify gradings, especially in finite types where cluster categories play a significant part.

On the one hand, gradings bring out some beautiful combinatorics, in the form of frieze patterns, but we will also consider some more algebraic applications of gradings. These include a "twisting" construction that is important for quantizing cluster algebra structures.

28th April 2014:
*Gradings on cluster algebras and associated combinatorics*

Pure Mathematics Colloquium, 4pm, Department of Mathematical Sciences, Durham University.

*For an abstract, please see my talk in Leicester below.*

27th March 2014:
*Clusters, combinatorics, grading, twisting and quantization*

London Algebra Colloquium, 5pm, School of Mathematical Sciences, Queen Mary, University of London.

*Abstract:*
Cluster algebras are currently playing a prominent role in several areas of mathematics, including Lie theory, representation theory and geometry (via mirror symmetry). We will introduce gradings on cluster algebras in a very natural way and show how to find and classify gradings, especially in finite types where cluster categories play a significant part.

On the one hand, gradings bring out some beautiful combinatorics, in the form of frieze patterns, but we will also consider some more algebraic applications of gradings. These include a "twisting" construction that is important for quantizing cluster algebra structures.

7th February 2014:
*Graded cluster algebras and quantum Grassmannians*

"Cluster algebras and combinatorics", Mathematical Institute, University of Münster, Germany.

*Abstract:*
I will introduce gradings on cluster algebras in a very natural way and show how to find and classify gradings, especially in finite types where cluster categories play a significant part. On the one hand, gradings bring out some beautiful combinatorics but we will also consider a more algebraic application of gradings as part of the proof that quantum Grassmannians are quantum cluster algebras.

5th November 2013:
*Gradings on cluster algebras and associated combinatorics*

Algebra-Geometry Seminar, 3pm, School of Mathematics and Statistics, Newcastle University.

*For an abstract, please see my talk in Leicester immediately below.*

8th October 2013:
*Gradings on cluster algebras and associated combinatorics*

Pure Mathematics Seminar, 2pm, Department of Mathematics, University of Leicester.

*Abstract:*
When studying any class of rings or algebras, the existence of a grading often has a big impact on what can be said about the members of the class. In the few years since their inception, cluster algebras have been found in numerous places and have been shown to be responsible for a plethora of combinatorial patterns, but until very recently gradings on cluster algebras have not been considered in any detail.

In this talk, we will introduce gradings on cluster algebras in a very natural way and show how the intricate structure and combinatorics associated to cluster algebras allows us to find and classify gradings. We will look at cluster algebras of finite type and examine the gradings they admit, making use of cluster categories. Conversely, the gradings bring out some beautiful combinatorics of their own, as we will see. Finally, we will conclude with a more algebraic application of gradings to a problem of the existence of certain (quantum) cluster algebra structures.

September 11th-13th 2013:
*Graded cluster algebras*

"Mirror Symmetry and Cluster Algebras", School of Mathematics, University of Leeds.

*Poster:* leeds13.pdf

August 17th 2012:
*A quantum analogue of a dihedral group action on Grassmannians*

XV International Conference on Representations of Algebras and Workshop (ICRA XV), Universität Bielefeld, Germany.

*Abstract:*
In recent work, Launois and Lenagan have shown how to construct a cocycle twisting of the quantum Grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the *n*-cycle *c*=(1 2 ... n), up to a power of *q*. This twisting is needed because *c* does not induce an automorphism of the quantum Grassmannian, as it does classically and semi-classically.

With Allman, we have described an extension of this construction to give a quantum analogue of the action on the Grassmannian of the dihedral subgroup of *S _{n}* generated by

November 30th 2011:
*Quantizing cluster algebras associated to coordinate rings *

Algebra seminar, Department of Mathematics, University of Glasgow.

*Abstract:*
I will give an introduction to quantum cluster algebras, using examples of coordinate rings and their quantizations to illustrate the definitions and the issues involved in establishing the existence of these structures. These examples include some of the most important quantum groups, such as quantum matrices, quantum *SL _{n}* and the quantum Grassmannian.

November 16th 2011:
*A dihedral group action on Grassmannians and its quantization*

Pure Mathematics seminar, Department of Mathematics and Statistics, Lancaster University.

*Abstract:*
The Grassmannian Gr(*k*,*n*) is the space of all *k*-dimensional subspaces of an *n*-dimensional space. This very nice geometric object has an action of the symmetric group *S _{n}* on it but this action does not respect some of the additional structure we might like to put on the Grassmannian. However a certain dihedral subgroup does and in this talk we will explore which structures this dihedral group preserves and also how this action can even be passed to the quantum version of the Grassmannian, but in a somewhat unexpected way.

October 22nd 2011:
*Some properties of quantum Grassmannians*

29th meeting of ARTIN (Algebra and Representation Theory in the North), School of Mathematics and Statistics, Newcastle University.

September 1st 2011:
*Quantum cluster algebras: a survey*

Kent Algebra Days 2011 (August 31st-September 2nd), School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury.

*Abstract:*
Quantum cluster algebras are a generalisation of the now ubiquitous cluster algebras, the latter being the commutative version of the theory. The non-commutativity in a quantum cluster algebra is relatively mild: the main feature is that elements in the same cluster must commute up to a power of *q*, although elements from different clusters may have more complicated relationships. The study of quantum cluster algebras has recently been boosted by the demonstration by Geiss, Leclerc and Schroer of quantum cluster algebra structures on the quantum coordinate rings of open cells of partial flag varieties, considerably extending the class of examples previously known. We will discuss the definition of quantum cluster algebras and some of their basic properties, and give an overview of what is known and unknown about these algebras at present.

May 11th 2011:
*The hitchhiker's guide to quantum algebra*

Kinderseminar, 11.30am, Room 2, Tom Gate, Christ Church College, Oxford.

May 3rd 2011:
*Some quantum analogues of properties of Grassmannians*

MAXIMALS seminar, 4.10pm, School of Mathematics, University of Edinburgh.

*Abstract:*
The classical coordinate ring of the Grassmannian has many nice structural properties and one expects these to carry over to its quantum analogue. We will discuss two properties for which this does indeed happen, namely a cluster algebra structure (recent work with Launois, quantizing work of Scott) and an action of the dihedral group (work with Allman, extending a recent construction of Launois and Lenagan). We will also mention an extension in a different direction, namely to infinite Grassmannians (work with Gratz).

February 15th 2011:
*A variety of flags*

Invariants Society meeting, 8.15pm, Mathematical Institute, Oxford.

*Abstract:*
Some mathematical ideas are both simple and pervasive but obliquely named, and flags are a good example of this. To a mathematician, a flag is a collection of subspaces of a vector space, each contained in the next and having dimension one less than its predecessor. So in three dimensions, we would picture a plane containing a line, both containing the origin. The collection of all flags has a fascinating geometrical and algebraic structure and we will explore some small examples and some related topics.

*Slides:* invariants11.pdf

November 12th 2010:
*Advanced counting: what do mathematicians do?*

Richardson Lecture, 5.30pm, Pusey Room, Keble College, Oxford.

*Abstract:* The chemist in a lab, the archaeologist in a trench, the historian in an archive, the mathematician... at a desk? Not all of the time of course, for any of these fields of study, but for many subjects we have some sense of where new understanding comes from and how it comes to be revealed. Research in mathematics, and especially in pure mathematics, is seen as mysterious by many but in fact it employs the same investigative tools as any other subject. Illustrated by a few of the lecturer’s own particular interests, we will explore how simple questions grow into grand theories, what the landscape of mathematics looks like and how it has been mapped by its many explorers. On a more mundane level, we will also find out what it is that mathematicians actually do all day - and yes, much of it is centered around that simple piece of furniture, the desk.

November 9th 2010:
*An introduction to cluster algebras and their quantum analogues*

Mathematics seminar, 2.30pm, Maths Lecture Theatre, Institute of Mathematics, Statistics & Actuarial Science, University of Kent.

October 19th 2010:
*Quantizing Grassmannians, Schubert cells and cluster algebras*

Algebra seminar, 4pm, Frank Adams Room 1, Alan Turing Building, School of Mathematics, University of Manchester.

*Slides:* manchester10.pdf

January 21st 2010:
*Quantizing Grassmannians, Schubert cells and cluster algebras*

Representation Theory seminar, 2.30pm, Mathematical Institute, Oxford.

*Slides:* oxford10.pdf

December 5th 2009:
*Quantizing Grassmannians, Schubert cells and cluster algebras*

2nd Belgian Mathematical Society - London Mathematical Society Conference, Katholieke Universiteit Leuven, Belgium.

*Abstract:* The quantum Grassmannians and their quantum Schubert cells are well-known and important examples in the study of quantum groups and quantum geometry. It has been known for some time that their classical counterparts admit cluster algebra structures, which are closely related to positivity properties. Recently we have shown that in the finite-type cases quantum Grassmannians admit quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky. We will describe these structures explicitly and also show that they naturally induce quantum cluster algebra structures on the quantum Schubert cells.

This is joint work with S. Launois.

*Slides:* leuven09.pdf

June 29th-July 3rd 2009:
*Examples of quantum cluster algebras associated to partial flag varieties*

"Geometric Representation Theory and Extended Affine Lie Algebras", University of Ottawa, ON, Canada.

*Poster:* Ottawa-2009-poster.pdf

6th May 2009:
*Partial flag varieties and quantum cluster algebras*

Algebra and Geometry seminar, 2.30pm, Department of Mathematics,
University of Bristol.

*Slides:* bristol09.pdf

31st March 2009:
*Partial flag varieties and quantum cluster algebras*

44th Bristol-Leicester-Oxford Colloquium (BLOC), 4pm, BEN
LT5, Department of Mathematics, University of Leicester.

*Slides:* 44thBLOC-web.pdf

23rd-27th March 2009:
*Braided enveloping algebras and quantum cluster algebras related to partial flag varieties*

"Algebraic Lie Structures with Origins in Physics", Isaac Newton Institute for Mathematical Sciences, Cambridge.

*Poster:* ALTW02-2009-poster.pdf

20th November 2008:
*Cluster algebras, partial flag varieties and quantum groups*

London Algebra Colloquium, 4.45pm, Room G2, School of Mathematical Sciences, Queen Mary, University of London.

1st April 2008:
*Making modules into algebras*

Mathematics research seminar, 5pm, Room C322, Centre for Mathematical Science, City University, London.

*Abstract:* Modules for associative or Lie algebras are just vector spaces acted on by the algebra. This means that we can take the product of an algebra element with a module element in a sensible way. Typically, one cannot take the product of two module elements, though.

However, in certain circumstances, it is possible to make a module into an associative or Lie algebra, in a way that is compatible with the action. I will start with some examples for finite-dimensional Lie algebras, then infinite-dimensional Kac-Moody Lie algebras and finally quantized enveloping algebras. The products we get will turn out to involve braidings, giving us braided Lie algebras and braided enveloping algebras. I will also demonstrate one use for this extra structure, namely gluing together the module and the original algebra to get bigger algebras.

*Slides:* kent-city-web.pdf

31st March 2008:
*Making modules into algebras*

Pure and Applied Mathematics seminar, 2.30pm, the McVittie Library, Institute of Mathematics, Statistics & Actuarial Science, University of Kent.

*(For an abstract and a PDF document containing the slides, see
my
talk in City
University.)*

11th October 2007:
*Some algebras associated to quantum parabolic subalgebras,*

Representation Theory seminar, 2.30pm, Mathematical Institute, Oxford.

24th September 2007:
*Some algebras associated to quantum parabolic subalgebras,*

Algebra seminar, 4.30pm, Roger Stevens Lecture Theatre 1, Department of Pure Mathematics, University of Leeds.

20th August 2007:
*On the inductive construction of quantized enveloping algebras,*

XII International Conference on Representations of Algebras and Workshop (ICRA XII), Nicolaus Copernicus University, Toruń, Poland.

*Abstract:* I will describe an inductive scheme for quantized enveloping algebras, arising from certain inclusions of the associated root data. These inclusions determine an algebra-subalgebra pair with the subalgebra also a quantized enveloping algebra: we want to understand the structure of the "difference" between the algebra and the subalgebra. Our point of view treats the background field and quantization parameter *q* as fixed and the parameter space as being the graph with vertices root data and edges given by their inclusions. More simply, one can think of the addition and deletion of nodes of Dynkin diagrams; we are interested in how the quantized enveloping algebras associated to the different diagrams are related.

By means of the Radford-Majid theorem, we see that associated to each inclusion there is a graded Hopf algebra in the braided category of modules of the subalgebra - this graded braided Hopf algebra is the object that describes the difference between the two algebras. Then using a construction of Majid called double-bosonisation, we can reconstruct the larger algebra from a central extension of the subalgebra, the graded Hopf algebra and its dual, generalising the usual triangular decomposition. I will focus on describing the structure of the graded braided Hopf algebra, showing that it is of a special kind, namely a Nichols algebra. I will also illustrate the theory with explicit examples, including the natural inclusion of *U _{q}(sl_{3})* in

*Presentation slides:* torun20070820.pdf

1st June 2006:
*Diagram algebras in glorious technicolor,*

QuIPS (Queen Mary Internal Postgraduate Seminar), 2pm, Room 103, School of Mathematical Sciences, Queen Mary, University of London.

10th November 2005:
*Inductive constructions for Lie bialgebras and Hopf algebras,*

Representation Theory seminar, 2.30pm, Mathematical Institute, Oxford.

23rd May 2005:
*Inductive constructions associated to Dynkin diagrams,*

Pure Mathematics seminar, 4.30pm, Room 103, School of Mathematical Sciences, Queen Mary, University of London.

*Abstract:* Dynkin diagrams appear in many disparate areas of mathematics but have their origins in Lie theory. First, I will describe two classes of objects associated to Dynkin diagrams, namely Lie bialgebras and quantized enveloping algebras. Although we may construct these directly for a given Dynkin diagram, I have considered an inductive approach. Dynkin diagrams are a particular class of graphs, which may be arranged into some infinite series together with some "exceptional" diagrams. We can consider adding vertices and edges to build up the diagrams. Via a very general construction, we may transfer this to the above associated objects and hence study their structure. I will give examples and some of the results I have obtained.

14th May 2005:
*Quantum Lie restriction,*

14th LMS Workshop on Quantum Groups and Noncommutative Geometry, School of Mathematical Sciences, Queen Mary, University of London.

22nd March 2005:
*Quantum Lie induction,*

Quantum Groups seminar, 4pm, Room 513, School of Mathematical Sciences, Queen Mary, University of London.

10th December 2004: *On Lie induction and the exceptional series,*

Séminaire sur les Algèbres Enveloppantes, 2pm, Institut de Máthematiques de Jussieu, Paris.

25th June 2004:
*Lie induction: constructing Lie bialgebras,*

"Representation Theory and its Applications", a satellite conference to the Fourth European Congress of Mathematics, Department of Mathematics, Uppsala University, Sweden.

*Abstract:* Lie bialgebras occur as the principal objects in the infinitesimalisation of the theory of quantum groups - the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In particular, a construction of Majid allows us to build Lie bialgebras following and moving between the Dynkin series, which we call Lie induction. We will see how this brings together the structure theory and the representation theory of Lie bialgebras.

A particular motivation for this work is to better understand the structure of the exceptional Lie algebras. We may also look to the general theory of Lie induction to try to see obstructions to the continuation of the *E*, *F* and *G* series - for example, the lack of a finite-type *E*_{9}.

20th January 2004:
*Lie bialgebras and Lie induction,*

Topology and Geometry seminar, 3pm, Colloquium Room A, Level 3, Mathematics Department building (S14), National University of Singapore.

*(See also my talk in Uppsala.)*

20th October 2003:
*Lie bialgebras: some inductive constructions,*

Quantum Groups seminar, 3pm, Room 513, School of Mathematical Sciences, Queen Mary, University of London.

9th October 2003:
*Too many axioms spoil the algebra...,*

QuIPS (Queen Mary Internal Postgraduate Seminar), 3.15pm, Room G2, School of Mathematical Sciences, Queen Mary, University of London.

*Abstract:* I will try to show how we can rewrite many algebraic notions in terms of maps and how this approach leads us very naturally to Hopf algebras. I hope to show how category theory plays an important role too, as more than just formalism. No previous knowledge of these topics will be assumed!

5th June 2003:
*An introduction to Lie algebras and Lie induction,*

QuIPS (Queen Mary Internal Postgraduate Seminar), 4pm, Room G2, School of Mathematical Sciences, Queen Mary, University of London.

*Abstract:* The first section of the talk will be an introduction to (some of) the theory of Lie algebras. We will mainly concentrate on their algebraic structure and in particular on the Dynkin diagram classification.

In the second part, we look at Lie induction, where we construct a Lie algebra from a sub-Lie algebra and a representation of the sub-Lie algebra. This is a "new from old" method to construct all (simple) Lie algebras from the very simplest and possibly most famous one, *sl*(2).

21st October 2002:
*Fuss-Catalan algebras: the Temperley-Lieb algebras in colour,*

Quantum Groups seminar, 3.30pm, Room 513, School of Mathematical Sciences, Queen Mary, University of London.