zbMATH — the first resource for mathematics

Nondegeneracy of the Lie algebra \(\text{aff}(n)\). (English) Zbl 1033.17023
The authors show that the Lie algebra of affine transformations \(\text{aff}(n)\) of \(\mathbb{R}^n\) is formally and analytically nondegenerate in the sense of A. Weinstein [ J. Differ. Geom. 18, 523–557 (1983; Zbl 0524.58011)]. It is an open problem to find and classify nondegenerate Lie algebras, and, up to now, very few nondegenerate Lie algebras are known. In this paper, the authors first explain and improve a new method (obtained in previous works) to linearize Poisson structures by first looking for a semi-linearization. Then, they prove the main result: the fact that \(\text{aff}(n)\) is formally and analytically nondegenerate. Finally, they show that this nondegeneracy unfortunately cannot be extended to \(e(n)\), the algebra of Euclidean transformations and \(\text{saff}(n)\), the algebra of volume preserving affine transformations. More precisely, it is proved that the main sisters of \(\text{aff}(n)\), that is \(e(2)\) and \(\text{saff}(3)\), are degenerate (and the same should be true for \(e(n)\) and \(\text{saff}(n)\) for all \(n\) as well).

17B63 Poisson algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
Full Text: DOI arXiv
[1] Arnold, V.I., Geometrical methods in the theory of ordinary differential equations, (1988), Springer-Verlag New York
[2] Conn, J.F., Normal forms for analytic Poisson structures, Ann. of math. (2), 119, 3, 577-601, (1984) · Zbl 0553.58004
[3] Conn, J.F., Normal forms for smooth Poisson structures, Ann. of math. (2), 121, 3, 565-593, (1985) · Zbl 0592.58025
[4] Dufour, J.-P., Linéarisation de certaines structures de Poisson, J. differential geom., 32, 2, 415-428, (1990) · Zbl 0728.58011
[5] Dufour, J.-P.; Molinier, J.-Ch., Une nouvelle famille d’algèbres de Lie non dégénérées, Indag. math. (N.S.), 6, 1, 67-82, (1995) · Zbl 0842.58023
[6] J.-C. Molinier, Linéarisation de structures de Poisson, Thèse, Montpellier, 1993
[7] Monnier, P.; Zung, N.T., Levi decomposition of smooth Poisson structures, preprint, 2002
[8] Wade, A., Normalisation formelle de structures de Poisson, C. R. acad. sci. Paris, Série I, 324, 5, 531-536, (1997) · Zbl 0890.58017
[9] Weinstein, A., The local structure of Poisson manifolds, J. differential geom., 18, 3, 523-557, (1983) · Zbl 0524.58011
[10] Zung, N.T., Levi decomposition of analytic Poisson structures and Lie algebroids, preprint, 2002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.