SQUARE - Structures for Quivers, Algebras and Representations (n. 2022S97PMY)
Type
PRIN 2022, D.D. n. 104 del 2 febbraio 2022
CUP
C53D23002540006
Principal Investigator
Lidia Angeleri (Università degli Studi di Verona)
Other research units
UO1 - Lidia ANGELERI - Università degli Studi di VERONA
UO2 - Alessandro ARDIZZONI - Università degli Studi di TORINO
UO3 - Giovanni CERULLIRELLI - Università degli Studi di ROMA "La Sapienza"
UO4 - Jorge Nuno DOS SANTOS VITORIA - Università degli Studi di PADOVA
Duration
28/09/2023 - 31/01/2026
Description
We propose a research project in representation theory of associative algebras which combines two complementary lines of investigation. On one hand we aim at establishing new connections between different theoretical aspects. On the other hand, we want to use these insights to achieve a new understanding of the representation theory of certain distinguished classes of algebras, which in turn will serve as case studies and will inform our intuition for more general phenomena.The first line of investigation will mainly deal with a k-algebra A which will often be assumed to be finite-dimensional over an algebraically closed field k and will thus be determined by a quiver with possible relations. Our goal is to understand the interplay between several combinatorial or topological structures associated with A, such as the lattice of torsion pairs in the category of A-modules, the lattice of ring epimorphisms with domain A, the Ziegler spectrum of A, and the wall and chamber structure induced by stability conditions over A.The notion of a torsion pair will play a central role in these investigations. In fact, our interest in this concept goes beyond the setting of abelian categories, and we will explore existence and constructions of torsion pairs in more general contexts. Another fundamental tool for our investigations will be silting theory, and in particular a new approach to mutation in triangulated categories via the concept of a large (i.e. not necessarily compact) cosilting object. We will further employ tilting theory to address some problems from algebraic geometry, including a conjecture by Bondal and Orlov.Further questions of a geometric nature will concern affine and projective varieties associated with a finite-dimensional algebra A. We plan to investigate the geometry of orbit closures inside the variety of A-modules of a fixed dimension and want to explore properties of quiver Grassmannians.The second line of investigation concerns the representation theory of specific classes of algebras. Wewill study path algebras of the form RQ where Q is a Dynkin quiver and R is a commutative noetherianring and expect to gain an insight in the structure of their derived categories. For a gentle algebra A, wewant to classify the cosilting objects and their mutations in the derived category of A and use thisknowledge to study the structure of the Bridgeland stability manifold.We further aim at a homological approach to the representation theory of Leavitt path algebras. As a first step towards this goal, we work at a description of their injective modules.We also intend to describe orbit closures of symmetric representations of symmetric quivers and explore applications in the theory of cluster algebras.Moreover, we consider bialgebras and use functorial techniques to address questions such as existence of integrals or Hopf structures. Finally, we study categorical properties of pointed Hopf algebras.
