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recap and outlook

After a lengthy spring-break, let us continue with our course on noncommutative geometry and SL2(Z)-representations. Last time, we have explained Grothendiecks mantra that all algebraic curves defined over number fields are contained in the profinite compactification
SL2(Z)^=lim SL2(Z)/N of the modular group SL2(Z) and in the knowledge of a certain subgroup G of its group of outer automorphisms
. In particular we have seen that many curves defined over the algebraic numbers Q correspond to permutation representations of SL2(Z). The profinite compactification SL2^=SL2(Z)^ is a continuous group, so it makes sense to consider its continuous n-dimensional representations repnc SL2^ Such representations are known to have a finite image in GLn(C) and therefore we get an embedding repnc SL2^repnss SLn(C) into all n-dimensional (semi-simple) representations of SL2(Z). We consider such semi-simple points as classical objects as they are determined by – curves defined over Q – representations of (sporadic) finite groups – modlart data of fusion rings in RCTF – etc… To get a feel for the distinction between these continuous representations of the cofinite completion and all representations, consider the case of Z^=lim Z/nZ. Its one-dimensional continuous representations are determined by roots of unity, whereas all one-dimensional (necessarily simple) representations of Z=C are determined by all elements of C. Hence, the image of rep1c Z^rep1 C is contained in the unit circle

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 SL2(Z) we can define its modular content (which will be the noncommutative tangent space in this classical point to the noncommutative manifold of SL2(Z)). In this way we will associate noncommutative gadgets to our classical object (such as orders in central simple algebras, infinite dimensional Lie algebras, noncommutative potentials etc. etc.) which give us new tools to study these objects. (2) conversely, as we control the tangentspaces in these special points, they will allow us to determine other SL2(Z)-representations and as we vary over all classical objects, we hope to get ALL finite dimensional modular representations. I agree this may all sound rather vague, so let me give one example we will work out in full detail later on. Remember that one can reconstruct the sporadic simple Mathieu group M24 from the dessin d’enfant

This
dessin determines a 24-dimensional permutation representation (of
M24 as well of SL2(Z)) 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
SL2(Z) 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 M24, 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 M24 we obtain representants
of all of this one-parameter family of simple
SL2(Z)-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 SL2(Z)? and what is its connection with the Kontsevich-Soibelman coalgebra? (12)
is there a noncommutative compactification of SL2(Z)? (and other arithmetical groups) (13) : how does one calculate the noncommutative curves associated to the Mathieu groups? (14) : whatever comes next… (if anything).

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