Before the vacation I finished a rewrite of the One quiver to rule them all note. The main point of that note was to associate to any qurve $A$ (formerly known as a quasi-free algebra in the terminology of Cuntz-Quillen or a formally smooth algebra in the terminology of Kontsevich-Rosenberg) a quiver $Q(A)$ and a dimension vector $\alphaA$ such that $A$ is etale isomorphic (in a yet to be defined non-commutative etale toplogy) to a ring Morita equivalent to the path algebra $lQ(A)$ where the Morita setting is determined by the dimension vector $\alphaA$. These “one-quiver settings” are easy to work out for a group algebra $lG$ if $G$ is the amalgamated free product of finite groups $G = H1 \bigstarH H2$.

Here is how to do this : construct a bipartite quiver with the left vertices corresponding to the irreducible representations of $H1$, say ${ S1, .. ,Sk }$ of dimensions $(d1, .. ,dk)$ and the right vertices corresponding to the irreducible representations of $H2$, ${ T1, .. ,Tl }$ of dimensions $(e1, .. ,el)$. The number of arrows from the $i$-th left vertex to the $j$-th right vertex is given by the dimension of $HomH(Si,Tj)$ This is the quiver I call the Zariski quiver for $G$ as the finite dimensional $G$-representations correspond to $\theta$-semistable representations of this quiver for the stability structure $\theta=(d1, .. ,dk ; -e1, .. ,-el)$. The one-quiver $Q(G)$ has vertices corresponding to the minimal $\theta$-stable dimension vectors (say $\alpha,\beta, .. $of the Zariski quiver and with the number of arrows between two such vertices determined by $\delta{\alpha \beta}-\chi(\alpha,\beta)$ where $\chi$ is the Euler form of the Zariski quiver. In the old note I've included the example of the projective modular group $PSL2(Z) = Z2 \bigstar Z3$ (which can easily be generalized to the modular group $SL2(Z) = Z4 \bigstar{Z2} Z6$) which turns out to be the double of the extended Dynkin quiver $\tilde{A5}$. In the rewrite I've also included an example of a congruence subgroup $\Gamma0(2) = Z4 \bigstar{Z2}^{HNN}$ which is an HNN-extension. These are somehow the classical examples of interesting amalgamated (HNN) groups and one would like to have plenty of other interesting examples. Yesterday I read a paper by Karen Vogtmann called Automorphisms of free groups and outer space in which I encountered an amalgamated product decomposition for $GL2(Z) = D8 \bigstar{Z2 \times Z2} (S3 \times Z2)$where $D8$ is the diheder group of 8 elements. When I got back from vacation I found a reference to this result in my mail-box from Warren Dicks. Theorem 23.1, p. 82, in Heiner Zieschang, Finite Groups of Mapping Classes of Surfaces, LNM 875, Springer, Berlin, 1981.

I worked out the one-quiver and it has the somewhat strange form depicted above. It is perfectly possible that I made mistakes so if you find another result, please let me know.

added material (febr 2007) : mistakes were made and the correct one quiver can be found elsewhere on this blog.

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Posted in geometry
Written on Thu, 26 August 2004 at 10:09 am
Tags: arxiv, Cuntz, groups, Kontsevich, modular, non-commutative, Quillen, representations
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