### Algebraic and geometric aspects of Lie theory

Project Type: Prin 2022

Funded by: MUR

Period: Sep 28, 2023 – Sep 27, 2025

Budget: €49.650,00

Principal Investigator: Alberto De Sole (Sapienza Università di Roma)

Local coordinator: Mario Salvetti (Università di Pisa)

##### Participants

Rocco Chirivì (Università del Salento), Giovanni Gaiffi (Università di Pisa), Andrea Maffei (Università di Pisa)

##### Description

The framework of the present project is the field of Lie Theory, which can be understood as the study of “symmetries” and has links and applications in several directions, both in mathematics and in theoretical physics. The extremely diverse nature of problems, techniques and perspectives it involves is reflected in the project, which comprises mathematicians of different background and specialisation.
For the sake of convenience, we will split the description of the project in three main sections.
> VERTEX ALGEBRAS AND W-ALGEBRAS. We investigate the structure, representations and applications of some algebraic objects whose definition originates in theoretical physics. Many of these structures are connected with the notion of vertex algebra, with special focus on a family of interesting examples: W-algebras.
> HYPERPLANE ARRANGEMENTS AND COXETER GROUPS. Here, we deal with the combinatorial investigation of hyperplane and toric arrangements and the description of the so-called Kazhdan-Lusztig cells. Particularly relevant in this section is the K($\pi$,1) conjecture.
> GEOMETRIC METHODS IN REPRESENTATION THEORY. We study some geometric structures that are of interest in representation theory. We will present some classical problems that deal with nilpotent orbits, standard monomial theory and homogeneous varieties, and a geometric approach towards Nichols algebras.
Before the very description of the project, we would like to stress that the above splitting is only a rhetorical expedient, rather than a concrete breakdown of what is instead a unitary research plan.
Combinatorics of root systems and Coxeter groups is a universal tool in the research we are about to outline and serves as a lingua franca for the entire project. Reduction of topological, geometric or representation theoretical problems to combinatorial structures constitutes one of the paradigms of Lie theory. Thes facts transpires directly also from our research interests: in the study of hyperplane arrangements, in the classification problems, in the investigation of coordinate rings. The use of symmetries in the study and construction of geometric structures sits at the core of the outmost developments in Lie Theory. On the other hand, reformulating Lie theoretical problems in a geometric language is a common technique toward their solution. Many results in W-algebra theory, for instance, directly translate into corresponding claims in the geometry of nilpotent orbits.
As a result, problems that are native to a certain area may find their solution in a different one, and even shed light on apparently distant questions. Many of the members of the present projects enjoy long-standing collaborations and there is a constant exchange of researchers and students among the research nodes. We are therefore convinced that the interaction between Lie theorists with diverse formations and backgrounds is of paramount importance for achieving a team’s research interests.