56th ARTIN Meeting
Algebra and Representation Theory
in the North
Nov. 28-29, 2019
University of Edinburgh
List of Speakers
Celeste Damiani (University of Leeds)
Maxime Fairon (University of Glasow)
João Faria Martins (University of Leeds)
Anna Felikson (University of Durham)
Iain Gordon (University of Edinburgh)
Sira Gratz (University of Glasgow)
Evgeny Sklyanin (University of York)
Robert Weston (Heriot-Watt University)
Makoto Yamashita (University of Oslo)
Charles Young (University of Hertfordshire)
Daniele Valeri (Daniele.Valeri@glasgow.ac.uk)
Venue: Bayes Centre (5th floor) [Google Maps]
Registration: if you wish to participate, please fill this form
Schedule of the event
(Almost) all talks will be in the Lecture Theatre 5.10.
Talks marked by (*) will be in the Seminar Room 5.02.
||Thursday, Nov. 28
||Friday, Nov. 29
Titles and Abstracts
Speaker: Celeste Damiani (University of Leeds)
Title: Loop braid groups and a new lift of Artin's representation
Artin’s representation for braid groups lifts to Dahm’s homomorphism in the case of
(extended) loop braid groups. We give by an injection of the (extended) loop braid group into the group
of automorphisms of a triple, composed by a group, an abelian group, and an action of the first on the
second one. We show this injection is both an extension of Artin’s representation for braid groups and of
Dahm’s homomorphism for (extended) loop braid groups (joint with J. Faria Martins and P. Martin).
Speaker: Maxime Fairon (University of Glasgow)
Title: What is... the elliptic Calogero-Moser space?
The Calogero-Moser (CM) system is an integrable system describing the
classical motion of particles on a line which are subject to a potential
of interaction that is either rational, trigonometric/hyperbolic or elliptic.
With the exception of the elliptic case, it is well-known that the phase space
of the CM system can be obtained by Hamiltonian reduction of a suitable space
of matrices. It is then a natural question to ask if such a reduction picture
also exists in the elliptic case if we start with a finite-dimensional space of
matrices. To tackle this problem in the complex setting, I will interpret the
known phase spaces in terms of specific non-commutative algebras. Moreover,
I will translate the Poisson structure of these phase spaces on the latter
algebras using a version of non-commutative Poisson geometry introduced by Van
den Bergh. I will then explain what is the natural elliptic analogue of these
non-commutative algebras, and how this leads to the elliptic CM space. This is
based on joint work with Oleg Chalykh (Leeds).
Speaker: Joao Faria Martins (University of Leeds)
Title: Crossed modules, homotopy 2-types, knotted surfaces and loop braids
I will review the construction of invariants of knots, loop braids and knotted surfaces
derived from finite crossed modules. I will also show a method to calculate the algebraic
homotopy 2-type of the complement of a knotted surface $\Sigma$ embedded in the 4-sphere
from a movie presentation of $\Sigma$. This will entail a categorified form of the
Wirtinger relations for a knot group.
Speaker: Anna Felikson (University of Durham)
Title: Quiver mutations, reflection groups and curves on punctured disc
Mutations of quivers were introduced by Fomin and Zelevinsky in 2002
in the context of cluster algebras. For some classes of quivers,
mutations can be realised using geometric or combinatorial models. We
will discuss a construction of a geometric model for all acyclic
quivers. The construction is based on the geometry of reflection
groups acting in quadratic spaces. As an application, we show an easy
and explicit way to characterise real Schur roots (i.e. dimension
vectors of indecomposable rigid representations of Q over the path
algebra kQ), which proves a recent conjecture of K.-H. Lee and K. Lee
for a large class of acyclic quivers. This work is joint with Pavel
Speaker: Iain Gordon (University of Edinburgh)
Title: Calogero-Moser cells and the Robinson-Schensted algorithm
This is joint work with Adrien Brochier and Noah White. I will explain a construction of Bonnafe-Rouquier about a partition of the elements of a finite Coxeter group which arises from the theory of rational Cherednik algebras. In the case of the symmetric group, this construction is related to the Galois theory of the (rational) Calogero-Moser phase space. Building on work of Aguirre-Felder-Veselov and of Halacheva-Kamnitzer-Rybnikov-Weekes, I will show that the construction yields a continuous version of the Robinson-Schensted algorithm and as a result confirms a conjecture of Bonnafe-Rouquier.
Speaker: Evgeny Sklyanin (University of York)
Title: Gaudin's infinite-dimensional R-matrix and quantum Kadomtsev-Petviashvili (KP) equation
In 1988, Michel Gaudin has found a peculiar infinite-dimensional R-matrix that,
besides quantum Yang-Baxter equation (YBE) and classical YBE, satisfies also
the so-called associative YBE from which the former two follow. We show that
Gaudin's R-matrix underlies the integrability of quantum KP and KdV equations.
We discuss also the Gauss (triangular) decomposition of the R-matrix and its
relation to the algebraic Bethe ansatz.
This is ongoing work with Alessandro Torrielli (Surrey).
Speaker: Robert Weston (Heriot-Watt University)
Title: Baxter's Q Operator for Open Spin Chains
I will first review the purpose and algebraic construction of Baxter's
Q-operator for closed quantum spin chains. I will then move on to a
parallel discussion for closed spin chains; here the algebraic
construction is more subtle due to the four different algebras that are
in play (the quantum group, two Borel subalgebras and a coideal
subalgebra). I will navigate this sea of algebras and give a new
construction of the Q-operator for the open XXZ chain with diagonal
Speaker: Makoto Yamashita (University of Oslo)
Title: Categorical quantization of symmetric spaces and reflection equation
Reflection equation is a powerful guiding principle to quantize Poisson homogenous spaces
into actions of quantum groups. I will explain a universality property of module categories
from cyclotomic Knizhnik-Zamolodchikov equations in the formal `multiplier’ algebra setting,
where the reflection operator becomes a complete invariant of categorical structure. As an
application we obtain a Kohno-Drinfeld type result comparing the type B braid group representations
from the KZ equations on the one hand, and the universal R-matrix and the universal K-matrix of
Balagovic and Kolb on the other. The proof relies on elementary but curious combination of Lie
algebra cohomology, Hochschild cohomology, and formality machinary from noncommutative geometry
inspired by works of Calaque and Brochier. Based on joint works with Kenny De Commer, Sergey Neshveyv,
and Lars Tuset.
Speaker: Charles Young (University of Hertfordshire)
Title: Affine opers and hypergeometric integrals of motion
To any Kac-Moody algebra, one can associate a quantum Gaudin model. For algebras of finite type,
the Bethe ansatz for these models has yielded deep results including isomorphisms between the
commutative algebra of Hamiltonians (called the Bethe algebra) and algebras of functions on spaces
of opers for the Langlands dual algebra. One would like analogous results for affine-type Gaudin
models, since they are closely related to integrable quantum field theories. I will discuss opers
in affine type, and show that the functions on spaces of affine opers take the form of certain
hypergeometric-type integrals. That leads to a conjecture that there exists a hierarchy of higher
affine Gaudin Hamiltonians also given by such integrals. I will describe this conjecture, and give
some supporting evidence coming from GKO-type coset constructions of the Virasoro and W3 algebras.
This talk is based on joint work with Sylvain Lacroix and Benoit Vicedo, in the papers arXiv:1804.01480,
arxiv:1804.06751 and work in progress.