Higgs spaces, loop crystals and representation of loop Lie algebras

Lead Research Organisation: University of Edinburgh
Department Name: Sch of Mathematics


The notion of group comes from the consideration of the set of symmetries of a given object. Conversely, given a group, we can ask which objects have a set of symmetries corresponding to this group. Such an object is called a representation of the group, and representation theory is about solving the problem of finding all these representations.My work concerns geometric representation theory. Namely, I am interested in constructing algebraic objects such as Lie algebras and algebraic groups in terms of convolution algebras of functions on geometric objects. This method has proven to be very fruitful in the 90s, when many combinatorial objects associated to groups and their representations, such as characters, were interpreted in terms of geometric invariants of some varieties. They were then used to prove several important conjectures.The main purpose of my research is to introduce these kind of results to a new set of algebras called loop Lie algebras, and to relate them to another set of geometric objects called Higgs fields. A new combinatorial object, which I call a loop crystal, should be the crucial link between the algebraic and geometric parts. This loop crystal, which I have already constructed in the simplest possible case, should lead to a new approach to conjectures in geometry. Conversely, this should provide powerful new tools to study representation theory.All these results have many connections to other flourishing domains such as cluster algebras, and is part of the Langlands Program philosophy, which involves a lot a different areas of mathematics, from geometry to number theory.

Planned Impact

The main beneficiaries of my research will be other academics. As my work lies inside pure mathematics, the direct beneficiaries would be mathematicians interested in geometric representation theory. But this area has many connections in large domains such as geometry, algebra and combinatorics. As such, my research will be profitable to the different workgroups in Edinburgh, both in algebra and geometry. It has also many interactions with the main topics studied in the ARTIN group or in the ICMS workshops. More generally many places in UK have reaserchers whose fields are closely related, as the University of Leeds, the University of Oxford or the Imperial College. More widely, many ideas in geometric representation theory come from physics (statistical mechanics or conformal field theory), and one should expect many results in this field to have consequences in physics in the longer term. I also plan to intend to many international conferences, and take every opportunity to share my work with other mathematicians. I will publish articles in internationally peer-reviewed journals, and give public access to my papers on arXiv, in order to interact with a broader audience.


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Description 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 bases.
Exploitation Route My findings can be separated in three parts. The first part, which is the original direction of the project, can be seen as the first step toward a theory of canonical bases, well-understood for Kac-Moody algebras, for some generalizations called the loop Kac-Moody algebras. It also leads to new tools needed for the description of moduli spaces of Higgs bundles on curves. These result would be useful for experts in these two fields. The second section is the result of my collaboration with Van Roosmalen and Liu, which is a generalization of the study of Hall algebras for large categories, which puts a new light on the so-called locally finite Lie algebras, for which little is known. It is expected that these algebras can be better understood via these results. The third part is my collaboration with Chlouveraki: the subject is the study of the so-called Yokonuma-Hecke algebras, which recently attracts a lot of attention, in view of the links with knot theory, as well as with complex reflection groups. They are for instance expected to produce new knot invariants for weighted knots.
Sectors Other

Description 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.
First Year Of Impact 2015
Sector Other
Impact Types Cultural

Description Hall algebras of directed categories 
Organisation Bielefeld University
Country Germany 
Sector Academic/University 
PI Contribution I used my expertise in Hall algebra to study the Hall algebras of interesting categories unraveled by Van Roosmalen: the directed categories.
Collaborator Contribution Van Roosmalen brought his expertise in directed categories, and Liu her expertise on locally finite Lie algebras.
Impact We have started to write an article together, which is still a work in progress, under the title: Hall algebras of directed categories.
Start Year 2013
Description Representation theory of Yokonuma-Hecke algebras 
Organisation Versailles Saint-Quentin-en-Yvelines University
Country France 
Sector Academic/University 
PI Contribution I worked with M. Chlouveraki on the sudy of the representation of Yokonuma-Hecke algebras.
Collaborator Contribution M. Chlouveraki told me about this project and we worked on it.
Impact One publication: Determination of representations and a basis for the Yokonuma-Temperley-Lieb algebra, with M. Chlouveraki, to appear on Algebras and Representation Theory.
Start Year 2013