Hall algebras

Lectures on Hall Algebras

These notes represent the written, expanded and improved version of a series of lectures given at the winter school “Representation theory and related topics” held at the ICTP in Trieste in January 2006. The topic for the lectures was “Hall algebras” and I have tried to give a survey of what I believe are the most fundamental results and examples in this area. The material was divided into five...

Algebras in Quantum Hall Theory

We show that U(∞) symmetry transformations of the noncommutative field theory in the Moyal space are generated by a combination of twoW1+∞ algebras in the Landau problem. Geometrical meaning of this infinite symmetry is illustrated by examining the transformations of an invariant subgroup on the noncommutative solitons, which generate deformations and boosts of solitons. In particular, we find ...

Hall Algebras as Hopf Objects

One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category Vect. In the past this problem has been resolved by working with a weaker structure called a ‘twisted’ bialgebra. In this paper we solve the problem differently by first switching to a d...

Canonical Bases of Singularity Ringel-hall Algebras and Hall Polynomials

In this paper, the singularity Ringel-Hall algebras are defined. A new class of perverse sheaves are shown to have purity property. The canonical bases of singularity RingelHall algebras are constructed. As an application, the existence of Hall polynomials in the tame quiver algebras is proved.

On Ringel-hall Algebras of Tame Hereditary Algebras