Project: Research

- Pouchin, Guillaume (Principal Investigator)

Status | Finished |
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Effective start/end date | 1/09/11 → 31/08/14 |

Total award | £290,514.00 |
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Funding organisation | EPSRC |

Funder project reference | EP/102610X/1 |

Period | 1/09/11 → 31/08/14 |
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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

bases.

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.

## Determination of the Representations and a Basis for the Yokonuma–Temperley–Lieb Algebra

Research output: Contribution to journal › Article

## Higgs bundles on weighted projective lines and loop crystals

Research output: Working paper