I have

been posting before on the necklace Lie algebra : on Travis

Schedler's extension of the Lie algebra structure to a Lie bialgebra

and its deformation and more recently

in connection with Michel Van den Bergh's double Poisson paper.

Yesterday, Victor Ginzburg and Travis Schedler posted their paper Moyal quantization of

necklace Lie algebras on the arXiv in which they give a Moyal-type

construction of the Hopf algebra deformation of the necklace Lie

bialgebra found by Schedler last year.

It would be nice if

someone worked out a few examples of these constructions in full detail.

But as often in the case of (wild) quiver situation it is not clear what

an 'interesting' example might be. For the finite and tame case

we have a full classification by (extended) Dynkin diagrams so a natural

class of examples but it isn't clear how to find gems in the

complement.

One natural source of double quiver situations seems

to come from what I called the One Quiver of a

formally smooth algebra. This one quiver of group algebras of some

interesting arithemetical groups such as the modular group

$PSL_2(\mathbb{Z}) $ and $SL_2(\mathbb{Z}) $ were calculated before and

turned out to be consisting of one (resp. two) components which are the

double of the tame quiver $\tilde{A}_5 $.

To obtain the double of

a wild quiver situation loook at the group $GL_2(\mathbb{Z}) = D_4

\bigstar_{D_2} D_6 $. In a previous post

I thought to have calculated it, but lately I found that this was

incorrect. Even the version I computed last week still had some mistakes

as Raf

Bocklandt discovered. But as of yesterday we are pretty certain that

the one quiver for $GL_2(\mathbb{Z}) $ consists of two components. One of

these is the double quiver of an interesting wild quiver

$\xymatrix{& \vtx{} \ar@{=}[rr] \ar@{=}[dd] & & \vtx{} \ar@{=}[dd]

\\ \vtx{} \ar@{=}[ur] \ar@{=}[rr] \ar@{=}[dd] & & \vtx{} \ar@{.}[ur]

\ar@{.}[dd] \ar@{=}[dr] \\ & \vtx{} \ar@{.}[rr] \ar@{=}[dr] & & \vtx{}

\\ \vtx{} \ar@{=}[rr] \ar@{.}[ur] & & \vtx{} \ar@{=}[ur]} $

where each double line indicates that there is an arrow in each

direction between the vertices. So, it is an interwoven pattern of one

big cycle of length 6 (reminiscent of the modular group case) with 4

cycles of length 5. Perhaps the associated necklace Lie (bi)algebra and

its deformation might be interesting to work out.

However, the

second component of the one quiver for $GL_2(\mathbb{Z}) $ is _not_

symmetric.Maybe I will come back to the calculation of these quivers

later.

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