Tomorrow

I’ll start with the course *Projects in non-commutative geometry*

in our masterclass. The idea of this course (and its companion

*Projects in non-commutative algebra* run by Fred Van Oystaeyen) is

that students should make a small (original if possible) work, that may

eventually lead to a publication.

At this moment the students

have seen the following : definition and examples of quasi-free algebras

(aka formally smooth algebras, non-commutative curves), their

representation varieties, their connected component semigroup and the

Euler-form on it. Last week, Markus Reineke used all this in his mini-course

Rational points of varieties associated to quasi-free

algebras. In it, Markus gave a method to compute (at least in

principle) the number of points of the *non-commutative Hilbert
scheme* and the varieties of

*simple representations*over a

finite field. Here, in principle means that Markus demands a lot of

knowledge in advance : the number of points of all connected components

of all representation schemes of the algebra as well as of its scalar

extensions over finite field extensions, together with the action of the

Galois group on them … Sadly, I do not know too many examples were all

this information is known (apart from path algebras of quivers).

Therefore, it seems like a good idea to run through Markus’

calculations in some specific examples were I think one can get all this

:

*free products of semi-simple algebras*. The motivating examples

being the groupalgebra of the (projective)

*modular group*

**PSL(2,Z) = Z(2) * Z(3)**and the free matrix-products $M(n,F_q) *

M(m,F_q)$. I will explain how one begins to compute things in these

examples and will also explain how to get the One

quiver to rule them all in these cases. It would be interesting to

compare the calculations we will find with those corresponding to the

path algebra of this

*one quiver*.

As Markus set the good

example of writing out his notes and posting them, I will try to do the

same for my previous two sessions on quasi-free algebras over the next

couple of weeks.