### Description

This project sits in the framework of Lie Theory and aims at developing the research guidelines of the young italian mathematicians working in this field.

Through the past century, Lie theory has developed in several directions, finding important applications in other areas of mathematics and physics. Its richness and perspectives are reflected in the different abilities of researchers participating in this project. We may summarize the main interest of this project in four themes:

* Infinite-dimensional Lie algebras, their generalizations and applications.

* Homogeneous varieties and their compactifications

* Topology of hyperplane arrangements and representations of Artin groups

* Combinatorics of Coxeter groups and groups generated by pseudoreflections.

Before describing the problems we intend to investigate, we would like to stress how the above division is essentially artificial, as there exist deep connections among these areas. First of all, techniques and tool are often very similar. Secondly, some problems that arise in one direction naturally induce questions that find their solution in another context. The combinatorics of root systems and Coxeter groups, for instance, is one of the fundamental ingredients in Lie theory, and some problems not directly connected with this language eventually amount to combinatorial statements. This is the case, for instance, of the problem of classifying finite and infinite-dimensional Lie algebras, studying symmetric and spherical varieties, as well as topological properties of hyperplane arrangements. In the same vein, some of the most significant results in the combinatorics of Coxeter groups are handled via the geometry of the above varieties, or the representation theory of Lie algebras.

We believe that the collaboration between Lie theorists that have distinct education and interests may constitute a fundamental ingredient for our goals.

The theory of semisimple Lie algebras, their classification and the study of their representations, have been some of the outmost successes of the first half of the 20th century. Since the 60's, the theory of semisimple Lie algebras has been generalized in many directions. Vertex algebras have been introduced by Borcherds in the 80's, drawing from Conformal Field Theory in physics, in the study of representations of the Fischer monster group. Lie superalgebras, and more recently n-superalgebras, are a natural generalization of Lie algebras and have gained growing importance in mathematics and theoretical physics, in the last few decades. Finally, quantizations of the enveloping algebra of a Lie algebra, the so-called quantum groups, have been introduced by Drinfeld and Jimbo in the 80's as an important tool in the study of classical problems, and have found applications in distant areas of mathematics. All of the above structures constitute a fundamental part of our research, which deals both with their structure and their representations.

An important aspect of the project is the application of such structures towards problems in mathematical physics. In particular, we intend to use Poisson vertex algebras, introduced in a paper of the PI along with V. Kac, to deal with the important problem of classifying Hamiltonian structures and the corresponding completely integrable systems.

Lie theory is closely related to homogeneous varieties and their compactifications. This link allowed both important results about representations of groups and Kac-Moody algebras via geometric constructions and the classification and explicit descriptions of certain notable varieties.

In this project we aim to use the representation theory to study both general properties (like singularities, irriducibility and reduction) and explicit descriptions of the coordinate rings of some varieties finding equations and special bases, on the pattern of the Plucker equations and the standard monomial theory for the usual grassmannian and for the Schubert varieties, as far as possible.

We aim to focus on spherical varieties (in particular model varieties), the conjugacy classes (and their closure) in a reductive group, the affine grassmannians, the quiver grassmannian and the invariant Hilbert scheme. These are important classes of varieties naturally appearing in the study of various Lie theory problems or in the study of certain moduli spaces. In recent years there has been a lot of reasearch activity in these fields.

An important part of the present project concerns the study of the geometry, topology and combinatorics of hyperplane arrangements (or more in general subspace arrangements) considering in particular the property of minimality and asphericality. In the toric case we will be interested in particular in arrangements defined by a root system. Moreover we will study the combinatorial counterpart of a toric arrangement, that is the arithmetic matroid, with applications to the partition function, polytopes and graphs. We will study the geometry and combinatorics of real and complex De Concini-Procesi models, with special concern for the non-minimal case. We will investigate the combinatorics and topology of Coxeter groups and their generalizations, paying attention in particular to complex braid groups, their minimal classification and their homology. Moreover we will consider the cohomology of braid groups with coefficients in representations arising from geometrical constructions and their polynomial extensions, with special interest for topological interpretations and stabilization properties.

We want to study some aspects of the representation theory and of the combinatorics of complex reflection groups, and of the associated root systems, and of some other related families of groups. In particular we want to complete the classification of projective reflection groups admitting a generalized involution model and the construction of a Gelfand model for all involutory reflection groups. We also want to study enumerative properties of quasi-parabolic sets for Coxeter groups, and extend their definition and Hecke algebra representations to complex reflection groups. We also plan to study the root polytope of an irreducible root system, i.e. the convex hull of all the roots of the system, and we want to show how the properties of the root polytope mirror the properties of the associated algebraic structures.

The richness of Lie theory and the wealth of problems we want to investigate require the commitment and collaboration of mathematicians with diverse interests. From this perspective, one of the strengths of our project is the presence of a group of young mathematicians, brought up in the best Italian and foreign universities, that have already produced important research in the areas we intend to confront.

Further, participants to the project have a long international experience. This allows us to intensively collaborate with some of the outmost

mathematicians in our field.

Lastly, we would like to stress how all members of our research group are still in the most productive phase of their scientific career, and a large

group of extremely young researchers, among which many brilliant doctoral students and recent PhDs, actively partecipate in the project.

Through the past century, Lie theory has developed in several directions, finding important applications in other areas of mathematics and physics. Its richness and perspectives are reflected in the different abilities of researchers participating in this project. We may summarize the main interest of this project in four themes:

* Infinite-dimensional Lie algebras, their generalizations and applications.

* Homogeneous varieties and their compactifications

* Topology of hyperplane arrangements and representations of Artin groups

* Combinatorics of Coxeter groups and groups generated by pseudoreflections.

Before describing the problems we intend to investigate, we would like to stress how the above division is essentially artificial, as there exist deep connections among these areas. First of all, techniques and tool are often very similar. Secondly, some problems that arise in one direction naturally induce questions that find their solution in another context. The combinatorics of root systems and Coxeter groups, for instance, is one of the fundamental ingredients in Lie theory, and some problems not directly connected with this language eventually amount to combinatorial statements. This is the case, for instance, of the problem of classifying finite and infinite-dimensional Lie algebras, studying symmetric and spherical varieties, as well as topological properties of hyperplane arrangements. In the same vein, some of the most significant results in the combinatorics of Coxeter groups are handled via the geometry of the above varieties, or the representation theory of Lie algebras.

We believe that the collaboration between Lie theorists that have distinct education and interests may constitute a fundamental ingredient for our goals.

The theory of semisimple Lie algebras, their classification and the study of their representations, have been some of the outmost successes of the first half of the 20th century. Since the 60's, the theory of semisimple Lie algebras has been generalized in many directions. Vertex algebras have been introduced by Borcherds in the 80's, drawing from Conformal Field Theory in physics, in the study of representations of the Fischer monster group. Lie superalgebras, and more recently n-superalgebras, are a natural generalization of Lie algebras and have gained growing importance in mathematics and theoretical physics, in the last few decades. Finally, quantizations of the enveloping algebra of a Lie algebra, the so-called quantum groups, have been introduced by Drinfeld and Jimbo in the 80's as an important tool in the study of classical problems, and have found applications in distant areas of mathematics. All of the above structures constitute a fundamental part of our research, which deals both with their structure and their representations.

An important aspect of the project is the application of such structures towards problems in mathematical physics. In particular, we intend to use Poisson vertex algebras, introduced in a paper of the PI along with V. Kac, to deal with the important problem of classifying Hamiltonian structures and the corresponding completely integrable systems.

Lie theory is closely related to homogeneous varieties and their compactifications. This link allowed both important results about representations of groups and Kac-Moody algebras via geometric constructions and the classification and explicit descriptions of certain notable varieties.

In this project we aim to use the representation theory to study both general properties (like singularities, irriducibility and reduction) and explicit descriptions of the coordinate rings of some varieties finding equations and special bases, on the pattern of the Plucker equations and the standard monomial theory for the usual grassmannian and for the Schubert varieties, as far as possible.

We aim to focus on spherical varieties (in particular model varieties), the conjugacy classes (and their closure) in a reductive group, the affine grassmannians, the quiver grassmannian and the invariant Hilbert scheme. These are important classes of varieties naturally appearing in the study of various Lie theory problems or in the study of certain moduli spaces. In recent years there has been a lot of reasearch activity in these fields.

An important part of the present project concerns the study of the geometry, topology and combinatorics of hyperplane arrangements (or more in general subspace arrangements) considering in particular the property of minimality and asphericality. In the toric case we will be interested in particular in arrangements defined by a root system. Moreover we will study the combinatorial counterpart of a toric arrangement, that is the arithmetic matroid, with applications to the partition function, polytopes and graphs. We will study the geometry and combinatorics of real and complex De Concini-Procesi models, with special concern for the non-minimal case. We will investigate the combinatorics and topology of Coxeter groups and their generalizations, paying attention in particular to complex braid groups, their minimal classification and their homology. Moreover we will consider the cohomology of braid groups with coefficients in representations arising from geometrical constructions and their polynomial extensions, with special interest for topological interpretations and stabilization properties.

We want to study some aspects of the representation theory and of the combinatorics of complex reflection groups, and of the associated root systems, and of some other related families of groups. In particular we want to complete the classification of projective reflection groups admitting a generalized involution model and the construction of a Gelfand model for all involutory reflection groups. We also want to study enumerative properties of quasi-parabolic sets for Coxeter groups, and extend their definition and Hecke algebra representations to complex reflection groups. We also plan to study the root polytope of an irreducible root system, i.e. the convex hull of all the roots of the system, and we want to show how the properties of the root polytope mirror the properties of the associated algebraic structures.

The richness of Lie theory and the wealth of problems we want to investigate require the commitment and collaboration of mathematicians with diverse interests. From this perspective, one of the strengths of our project is the presence of a group of young mathematicians, brought up in the best Italian and foreign universities, that have already produced important research in the areas we intend to confront.

Further, participants to the project have a long international experience. This allows us to intensively collaborate with some of the outmost

mathematicians in our field.

Lastly, we would like to stress how all members of our research group are still in the most productive phase of their scientific career, and a large

group of extremely young researchers, among which many brilliant doctoral students and recent PhDs, actively partecipate in the project.

Status | Active |
---|---|

Effective start/end date | 1/1/12 → … |

### Funding

- MIUR - FIRB

### Fingerprint

Combinatorics

Coxeter group

Root system

Physics

Model

Arrangement of hyperplanes

Representation theory

Arrangement

Lie algebra

Topology

Spherical varieties

Complex reflection groups

Vertex algebra

Infinite-dimensional Lie algebras

Reflection group

Semisimple Lie algebra

Braid group

Grassmannian

Compactification

Polytope