Séminaires : Séminaire d'Algèbre

Quasi-hereditary algebras were introduced by Cline, Parshall and Scott as a tool to study highest weight theories which arise in the representation theories of semi-simple complex Lie algebras and reductive groups. Surprisingly, there are now many examples of such algebras, such as Schur algebras, algebras of global dimension at most two, incidence algebras and many more. 

A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In the early examples the partial order predated (and motivated) the theory, so the choice was clear. However, there are instances of quasi-hereditary algebras where there is no natural choice for the partial ordering and even if there is such a natural choice, one may wonder about all the possible orderings.

In this talk we will explain that all these choices for an algebra $A$ can be organized in a finite partial order which is in relation with the tilting theory of $A$. In a second part of the talk we will focus on the case where $A$ is the path algebra of a Dynkin quiver.