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Rigidity of \(H\)-type groups. (English) Zbl 0982.22013

It is shown that if the dimension of the center of the \(H\)-type group \(N\) is at least three, then the generalized contact mappings are in the automorphism group of a finite dimensional Lie algebra \(\mathbf g\). This algebra is formed by the infinitesimal generators of a local one parameter subgroup of generalized contact transformations. The rigidity of the \(H\)-type groups is defined as the property that the Lie algebra \(\mathbf g\) is finite dimensional. It has been shown earlier that in the case of the complexified Heisenberg group, when the dimension of the center of the group \(N\) is two, the algebra \(\mathbf g\) is infinite dimensional. In the paper under review, the approach is based on the study of vector fields which generate local one parameter flows of contact diffeomorphisms. The first derivatives of these vector fields \(u\in{\mathbf g}\) can be pointwise interpreted as elements of the Lie algebra \(\text{aut}(N)\) of the group \(\text{Aut}(N)\) of grading preserving Lie algebra automorphisms. Then, by using the integrability conditions, one can ensure that \(u\) can only vary polynomially. Therefore, \(u\in {\mathbf g}\) has polynomial coefficients, and, consequently, \(\mathbf g\) is finite dimensional.
Reviewer: A.A.Bogush (Minsk)

MSC:

22E25 Nilpotent and solvable Lie groups
53C24 Rigidity results
22D45 Automorphism groups of locally compact groups
57R50 Differential topological aspects of diffeomorphisms
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