After a lengthy spring-break, let us continue with our course on noncommutative geometry and
and though these points are very special there are enough of them (technically, they form a Zariski dense subset of all representations). Our aim will be twofold : (1) when viewing a classical object as a representation of
- This
- dessin determines a 24-dimensional permutation representation (of
as well of ) which- decomposes as the direct sum of the trivial representation and a simple
- 23-dimensional representation. We will see that the noncommutative
- tangent space in a semi-simple representation of
is determined by a quiver (that is, an- oriented graph) on as many vertices as there are non-isomorphic simple
- components. In this special case we get the quiver on two points
- $\xymatrix{\vtx{} \ar@/^2ex/[rr] & & \vtx{} \ar@/^2ex/[ll]
- \ar@{=>}@(ur,dr)^{96} } $ with just one arrow in each direction
- between the vertices and 96 loops in the second vertex. To the
- experienced tangent space-reader this picture (and in particular that
- there is a unique cycle between the two vertices) tells the remarkable
- fact that there is **a distinguished one-parameter family of
- 24-dimensional simple modular representations degenerating to the
- permutation representation of the largest Mathieu-group**. Phrased
- differently, there is a specific noncommutative modular Riemann surface
- associated to
, which is a new object (at least as far - as I’m aware) associated to this most remarkable of sporadic groups.
- Conversely, from the matrix-representation of the 24-dimensional
- permutation representation of
we obtain representants - of all of this one-parameter family of simple
-representations to which we can then perform- noncommutative flow-tricks to get a Zariski dense set of all
- 24-dimensional simples lying in the same component. (Btw. there are
- also such noncommutative Riemann surfaces associated to the other
- sporadic Mathieu groups, though not to the other sporadics…) So this
- is what we will be doing in the upcoming posts (10) : explain what a
- noncommutative tangent space is and what it has to do with quivers (11)
- what is the noncommutative manifold of
? and what is its connection with the Kontsevich-Soibelman coalgebra? (12) - is there a noncommutative compactification of
? (and other arithmetical groups) (13) : how does one calculate the noncommutative curves associated to the Mathieu groups? (14) : whatever comes next… (if anything).
Comments