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?

This morning, Michel Van den Bergh posted an interesting paper on the arXiv entitled Double Poisson Algebras. His main motivation was the construction of a natural Poisson structure on quotient varieties of representations of deformed multiplicative preprojective algebras (introduced by Crawley-Boevey and Shaw in Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem) which he achieves by extending his double Poisson structure on the path algebra of the quiver to the 'obvious' universal localization, that is the one by inverting all 1+aa^{\star} for a an arrow and a^{\star} its double (the one in the other direction).
For me the more interesting fact of this paper is that his double bracket on the path algebra of a double quiver gives finer information than the necklace Lie algebra as defined in my (old) paper with Raf Bocklandt Necklace Lie algebras and noncommutative symplectic geometry. I will certainly come back to this later when I have more energy but just to wet your appetite let me point out that Michel calls a double bracket on an algebra A a bilinear map
\{ \{ -,- \} \}~:~A \times A \rightarrow A \otimes A
which is a derivation in the second argument (for the outer bimodulke structure on A) and satisfies
\{ \{ a,b \} \} = – \{ \{ b,a \} \}^o with ~(u \otimes v)^0 = v \otimes u
Given such a double bracket one can define an ordinary bracket (using standard Hopf-algebra notation)
\{ a,b \} = \sum \{ \{ a,b \} \}_{(1)} \{ \{ a,b \} \}_{(2)}
which makes A into a Loday algebra and induces a Lie algebra structure on A/[A,A]. He then goes on to define such a double bracket on the path algebra of a double quiver in such a way that the associated Lie structure above is the necklace Lie algebra.

Previous in series

Next in series

arxiv, geometry, necklace, noncommutative, representations