Multiplicative deformations of spectrale triples associated to left invariant metrics on Lie groups

Details Multiplicative deformations of spectrale triples associated to left invariant metrics on Lie groups Multiplicative deformations of spectrale triples associated to left invariant metrics on Lie groups

2009-06-16

arxiv.org/abs/0906.2887v1

Mathematics

Differential Geometry

23 pages

Scientific paper

We study the triple $(G,\pi,\prs)$ where $G$ is a connected and simply connected Lie group, $\pi$ and $\prs$ are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on $G$ such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the direction of $\pi$) of the spectrale triple associated to $\prs$ are satisfied. We show that the geometric problem of the classification of such triple $(G,\pi,\prs)$ is equivalent to an algebraic one. We solve this algebraic problem in low dimensions and we give the list of all $(G,\pi,\prs)$ satisfying Hawkins's conditions, up to dimension four.

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