Given a finite monoid G and its group algebra A over a field $K$. I have never really studied such algebras, but I have the following questions (which are probably basic questions on any large class of algebras):
Is it known when $A$ is selfinjective or a Hopf algebra?
More generally, what is known about the (finiteness of) Gorenstein (or global) dimension of $A$?
What is known about the dominant dimension of A? When is it larger than or equal to two (equivalently, when is A isomorphic to the endomorphism ring of a generator-cogenerator over some algebra)?
When is A isomorphic to a quiver algebra? When this is the case, is there a way via GAP/qpa to obtain the quiver with relations (in case the field is finite)? When is A connected? Is there a large class of such monoid algebras, where the quiver with relations is known?
When is A representation-finite?
Im also interested in partial results, for example by giving restrictions to G. Note that all questions have a well-known answer in case G is a group.
edit: 6. When is A local?
- For group algebras, one knows explicitly the algebras which appear as blocks of finite representation type (the Brauer tree algebras). Is something similar to be expected for general monoid algebras? That is: Can one describe the algebras appearing as finite representation type blocks of monoid algebras?
8.What is known about the homological conjectures for monoid algebras. For example: Is the Cartan-determinant always 1 in case the global dimension is finite? Is the left injective dimension of the regular module always equal to the right injective dimension of the regular module? (those conjectures are true for monomial algebras)