Nondegenerate Lie algebras
For a Poisson structure \(\{.,.\}\) on a manifold \(M\) we choose the local coordinates \((x_1,\ldots,x_n)\) vanishing at fixed \(x_0\in M\) and consider the Taylor expansion \(\{x_i,x_j\}=\sum_kC^k_{ij}x_k+\ldots\). Since \(\{.,.\}\) satisfies the Jacobi identity, the \(C_{ij}^k\) are structure constants of a well-defined Lie algebra \(g\) which is called the linear part of \(\{.,.\}\). A finite dimensional Lie algebra \(g\) is called \textit{nondegenerate} if for every Poisson structure which vanishes at the point \(x_0\in M\) and which has \(g\) as linear part, there exist new coordinates \((x_1',\ldots,x'_n)\) with \(\{x'_i,x'_j\}=\sum_kC^k_{ij}x'_k\). In addition to the ``nondegeneracy'' defined above, we can also consider formal nondegeneracy (where Poisson structures and changes of coordinates are formal) and analytic nondegeneracy (where Poisson structures and changes of coordinates are analytical). \textit{A. Weinstein} [J. Differ. Geom. 18, 523-557 (1983; Zbl 0524.58011)] proved that every semisimple Lie algebra is formally nondegenerate. \textit{J. F. Conn} proved in Ann. Math. 119, 577-601 (1984; Zbl 0553.58004) that every semisimple Lie algebra is analytically nondegenerate and in Ann. Math. 121, 565-593 (1985; Zbl 0592.58025) that every compact semisimple Lie algebra is \(C^\infty\)-nondegenerate. Let \(A_2\) be a nontrivial 2-dimensional Lie algebra. The main results of the paper are Proposition 1. If \(g\) is a semisimple Lie algebra, then the semidirect product \(g\propto\mathbb{R}\) is formally and analytically nondegenerate. If \(g\) is semisimple compact, then \(g\propto\mathbb{R}\) is \(C^{\infty}\)-nondegenerate. Proposition 2. The product \(A_2\propto\ldots\propto A_2\) of \(n\) copies of \(A_2\) is formally and analytically nondegenerate. \(A_2\propto A_2\) is \(C^\infty\)-nondegenerate.
scientific article; zbMATH DE number 1017600
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