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necklaces (again)

By lieven

non-commutative geometry

  1. Brauer’s forgotten group
  2. connected component coalgebra
  3. Galois and the Brauer group
  4. a noncommutative Grothendieck topology
  5. noncommutative geometry
  6. noncommutative geometry 2
  7. projects in noncommutative geometry
  8. points and lines
  9. more noncommutative manifolds
  10. the necklace Lie bialgebra
  11. the one quiver for GL(2,Z)
  12. representation spaces
  13. quiver representations
  14. moduli spaces
  15. cotangent bundles
  16. differential forms
  17. curvatures
  18. Brauer-Severi varieties
  19. smooth Brauer-Severis
  20. hyper-resolutions
  21. a cosmic Galois group
  22. double Poisson algebras
  23. A for aggregates
  24. B for bricks
  25. necklaces (again)
  26. seen this quiver?
  27. why nag? (2)
  28. why nag? (3)
  29. sexing up curves
  30. the Klein stack
  31. Alain Connes on everything
  32. noncommutative topology (1)
  33. a noncommutative topology 2
  34. noncommutative topology (3)
  35. noncommutative topology (4)
  36. non-geometry
  37. non-(commutative) geometry
  38. noncommutative Fourier transform
  39. noncommutative bookmarks
  40. noncommutative geometry : a medieval science?

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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Posted in geometry
Written on Wed, 23 March 2005 at 10:38 am
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