1. Key Findings
When I started the EPSRC Fellowship I was mainly working on the link
between loop Kac-Moody algebras and the moduli space of Higgs bundles
on weighted projective lines. In my PhD thesis I studied the simple case
of the complex projective line. I proved that a very natural algebra defined
in terms of constructible functions on the space of nilpotent Higgs bundles
is isomorphic to to a positive part of the enveloping algebra U(Âsl2), and
studied the irreducible components of this space to provide a special basis
for this algebra, as well as a combinatorial structure on this basis analogous
to the crystal structure for Kac-Moody algebras.
During my stay in Edinburgh I was able to generalise these constructions
to the case of general weighted projective lines. In this more general case
the situation is a lot more complicated, as the categories of objects involved,
coherent sheaves and Higgs bundles on these curves, do not have an explicit
and simple description compared to the case of the complex projective line.
I first proved in my article [P2] that the moduli space of nilpotent Higgs
bundles on any weighted projective line is pure (i.e. irreducible components
inside a connected component have the same dimension). The proof relies on
the construction of geometric correspondences between irreducible components,
in the spirit of the geometric construction of crystals of Kac-Moody
algebras, due to Kashiwara and Saito [KS]. The combinatorial structure
then obtained should be considered as an analog of a crystal, in the case
of loop Kac-Moody algebras. It consists of the data of set of irreducible
components, which I described in the simplest cases in my paper, together
with operators indexed by rigid indecomposable coherent sheaves. I called
this new combinatorial structure a loop crystal.
In an article in way of completion [P3], I study the Higgs algebra associated
to any weighted projective line. After describing general properties of these
algebras like the existence of a coproduct, I prove that in the case the genus
gX of the curve X is less or equal to 1, they are isomorphic to a corresponding
(the positive part of) a loop Kac-Moody enveloping algebra LgX. I then
produce a semicanonical basis for the Higgs algebra, parametrised by the
irreducible components. This basis inherits of the combinatorial structure of
[P2], which should describe a corresponding category of representation which
still needs to be defined. The restriction on the genus of the curve comes
from the use of the results of Schiffmann which are only conjectured in the
other cases. A more direct approach would allow to prove these conjectures.
Between January and March 2013, I stayed in the MSRI during a semester
on representation theory. During this stay I started two new collaborations
on different subjects.
First I started with Adam Van Roosmalen and Qunhua Liu a project on Hall
algebras of directed categories. The project relies on the description by Van
Roosmalen in [VR] of hereditary categories with Serre duality. It is a generalisation
of the classification, due to Happel, of hereditary categories with a
tilting object. In this case it is already known that the categories involved
are representations of quivers and coherent sheaves on weighted projective
lines. These categories have the right properties in order to have interesting
Hall algebras (i.e. quantum groups), and these algebras have already been
intensively studied (by Ringel, Lusztig, Schiffmann, see [L, Sc1, Sc2]). A new
class of categories appears in the classification of Van Roosmalen (which is a
generalisation of the classification of Happel): the categories of representations
of directed categories. In the article in preparation [LPV], we study the
structure of the Hall algebras of these categories and link with with infinite
versions of quantum enveloping algebras.
Secondly I worked with M. Chlouveraki on a new class of algebras called
Yokonuma-Temperley-Lieb algebras. The Yokonuma-Hecke algebras were
introduced by Yokonuma in [Yo], as generalisations of Hecke algebras. They
are defined in a similar way, as endomorphism algebras of the vector representation
of the general linear group over a finite field, commuting with
the action of a nilpotent subgroup (instead of a Borel subgroup for Hecke
algebras). They attracted a lot more attention during the last five years,
since the discovery of a nicer presentation (see [Ju1, Ju2, Ju3]), as well as
applications in knot theory (see [JuLa1, JuLa2, JuLa3]). In our article [CP1]
with Chlouveraki (to be published in Algebras and Representation Theory),
we describe the representation theory of the Yokonuma-Temperley-Lieb algebra,
an analog of Temperley-Lieb algebras for Yokonuma-Hecke algebras
introduced in [GoJuLa1], and describe an explicit basis of these.
In an article in preparation [CP2] with Chlouveraki, we describe the representation
theory of framizations of Temperley-Lieb algebras. These algebras,
introduced in [GoJuLa2], have expected applications in knot theory, in the
context of weighted braids and knots.
I also started to investigate the construction of the Yokonuma-Hecke algebras
geometrically, using functions on some nice spaces, in the spirit of my
previous work of my PhD thesis, published in [P1]. It should lead to natural
constructions such as Schur-Weyl dualities and the construction of particular
2. Activities and Impact
I had the pleasure to give talks and interact with many people during
my fellowship. I participated and contributed to several workgroups in Edinburgh
(perverse sheaves in representation theory, quiver Hecke algebras,
the 24 seminar). I also gave numerous talks in the United Kingdom (Glasgow,
Manchester, London, York), as well as in Europe, for seminars (Nancy,
Reims, Paris 7) and conferences (CIRM, Luminy).
I also strenghtened links with other universities through my collaborations:
first with the University of Versailles (Chlouveraki) and the people at University
of Athens (Lambropoulou and collaborators), which I visited and talked
with about our work on Yokonuma-Temperley-Lieb algebras and framizations
of Temperley-Lieb algebras. But also with Bielefeld and Prague: I
visited Van Roosmalen in Bielefeld, together with Liu, in order to progress
on our work on Hall algebras of directed categories.
I also visited Prof Lambropoulou at the National Technical University in
Athens, where we could discuss about the Yokonuma-Temperley-Lieb algebras.
A visit to Prof Juyumaya in Chile is also scheduled in a near future.