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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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##### References:
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