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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).

MSC:
17B63 Poisson algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
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