Homological questions on monoid algebras

Question

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):

1. Is it known when $A$ is selfinjective or a Hopf algebra?

2. More generally, what is known about the (finiteness of) Gorenstein (or global) dimension of $A$?

3. 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)?

4. 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?

5. 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?

7. 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)

Answer 1

These are a lot of questions. Let's work over complex numbers since the modular case is more complicated.

If the monoid M is (von Neumann) regular (each a=aba for some b) then CM is quasihereditary and so has finite global dimension. Monoid algebras can have infinite global dimension. If M is regular then self-injective is the same as semisimple for CM but in the non regular case it can be self-injective.

I characterized when a monoid has a basic algebra with Stuart Margolis, that is, when it is given by a quiver with relations. We give a method to reduce computing the quiver to group character theory but we don't know how to get relations in full generality. See http://arxiv.org/abs/1101.0416

Usually a monoid algebra is just a bialgebra and not hopf. I have a book coming out with lots on this.

I don't think there is much hope to classify representation finite monoids. It's pretty rare.

In the modular setting (the characteristic of the field divides the order of a maximal subgroup), then the algebra of a regular monoid has infinite global dimension and it is known when the algebra is self injective in this case.

Update As I commented on your related question, by the results of eudml.org/doc/143254 every connected representation finite algebra is a block in a monoid algebra.

Answer 2

About 1:

If $M$ is a monoid, then the bialgebra $B=k[M]$ (with comultiplication $\Delta(m)=m\otimes m$) is a Hopf algebra if and only if $M$ is actually a group. The proof is the followig:

The antipode $S:k[M]\to k[M]$ is a coalgebra map, so it send group-like to group-like. An easy exercise shows that the only group-like elements in $k[M]$ are the elements of $M$. Then $S(m)$ is necesarily the inverse of $m$, because the antipode property $\epsilon(h)=S(h_1)h_2=h_1S(h_2)$ means, for grouplikes, $1=S(m)m=mS(m)$.