The project is a continuation of previous ones with the same title and Principal Investigator. In this project the group of participants has expanded, so as to include all the structured staff of the Mathematics Department Guido Castelnuovo with algebraic research interests. The unification of previous smaller projects aims to improve mutual collaboration, to pool existing international collaborations, to create new ones and to carry out scientific events and support initiatives for young researchers more effectively. Unification is justified and motivated by the fact that many research topics fall into the comprehensive name of Representation Theory and have remarkable interest and applications in Mathematical Physics.
Diverse topics will be covered, all central to basic mathematical research and relevant to the fields of algebra, algebraic and complex geometry, topology, combinatorics (both algebraic and enumerative) and theoretical computer science. They range from representation theory of various kind of infinite-dimensional Lie algebras (affine algebras, vertex algebras, Lie superalgebras, Lie pseudoalgebras, cluster algebras) to more analytical and topological topics in the study of algebraic groups and complex geometry. Algebraic, combinatorial and theoretical computer science aspects will be covered, treated mainly via algebraic techniques. In detail, the topics studied, consistent with those of the previous editions of this and other projects, of which they are a refinement and deepening, will be the following:
- Unitarity in vertex operator algebras
- W-algebras and completely integrable Hamiltonian systems
- Lie pseudoalgebras
- Cluster algebras through Lie theory
- Spherical varieties
- Actions of complex algebraic and Lie groups
- Calibrations on manifolds
- Double quasi-symmetric functions
- PI Theory and related topics
- Integral geometry on trees
- Counting functions of languages
Unitarity in vertex operator algebras
The notion of unitarity is of fundamental importance, also in view of applications to Conformal Field Theory. The starting point is the existence of an invariant Hermitian form and it seems highly desirable to have a coherent general theory for a wide class of vertex algebras. Conformal vertex algebras seem a very promising candidate to develop the theory. This class of vertex algebras includes most of the relevant examples arising in Conformal Field Theory.
W-algebras and completely integrable Hamiltonian systems
The computation of cohomology of various families of vertex algebras and Poisson vertex algebras as well as the techniques developed for the calculation, that may divert from the semiclassical limit procedure, would have interesting applications to the integrability of important classes of Hamiltonian systems.
Lie pseudoalgebras
The Lie pseudoalgebraic language is mature and understood, and has attracted much attention in the last 10-20 years. Differential geometric interpretations of the representation theory of linearly compact Lie algebras has recently had a new impulse thanks to the generalization by Eastwood of the de Rham and Rumin constructions to locally conformally symplectic manifolds. Super-geometry generalizations and a possible analogy with projective representations of the Hamiltonian Lie algebra seem promising.
Cluster algebras through Lie theory
Since the inception of cluster algebras, describing their bases has been one of the main driving force behind their study. This has led to many different constructions tying in ideas from a variety of fields like Teichmüller theory [4], Auslander-Reiten theory [5], and mirror symmetry [6]. The aim of this project is to unify these constructions as already done in a special case [3].
Spherical varieties
A proof of Stanley's conjecture would allow to describe the support of the decomposition of the product of Jack polynomials and provide a very good understanding of the product of spherical functions in type A. We have a new strategy to prove part of the conjecture looking at skew Jack polynomials which may have integral coefficients and to generalize the conjecture for all root systems and consequently to a quite complete understanding of the product of Harish-Chandra spherical functions for all symmetric spaces.
Actions of complex algebraic and Lie groups
The combinatorial aspects of multiplicity-free manifolds have been used, e.g., by Knop in the proof of the Delzant conjecture, and those of spherical varieties by Luna, Brion, Knop and others in crucial ways. This research brings them closer together. New insight is also expected on how the local structure theorem for algebraic group actions relates to the local study of symplectic manifolds.
Calibrations on manifolds
A comparison of the differential forms generating the cohomology of exceptional symmetric spaces of compact type promises to be an innovative advance in the indicated topics. This includes the study of twistor spaces and other fibrations, as well as their curvature properties, developing a diagram well present in quaternion Kähler geometry and already studied for the Cayley planes F_4/Spin(9) and E_6/Spin(10)U(1).
Double quasi-symmetric functions
We will introduce a new Hopf algebra of functions in an infinity of variables, the algebra of double quasi-symmetric functions. This algebra will: contain the quasi-symmetric functions as a subalgebra; be on bidimensional variables, that is, can be thought of as a subalgebra of series in variables indexed in the partial order of NxN; better capture those aspects of double posets, double quasi-posets, Schur functions, etc.
PI Theory and related topics
The techniques developed for identities of *-superalgebras seem to be useful for superalgebras with superinvolution. Along the way, we want to investigate the question whether suitable subalgebras of upper block triangular matrix algebras can be determined by their *-graded identities, giving a contribution to the Isomorphism Problem in PI Theory. Finally, the classification of group rings with solvable unit group of small derived length extends classical results of Shalev to the case of non-torsion groups.
Integral geometry on trees
A possible approach is to impose suitable postulates on a topological space and to associated structures thereof that would ensure the possibility of it being the boundary of a tree and to devise a procedure to construct (or reconstruct) the geometric structure of the tree from this information.
Counting functions of languages
An innovative aspect is related to the connection between counting functions of languages and vector partition functions, which will make available the use of several tools of enumerative combinatorics of polyhedra and box-spline algebra and extend results that hold for algebraic languages to context-sensitive sets and their counting maps, for which little is known.