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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Distributions and the Lie algebras their bases can generate


Author: Henry Hermes
Journal: Proc. Amer. Math. Soc. 106 (1989), 555-565
MSC: Primary 58A30; Secondary 17B30, 53A55, 93B27
MathSciNet review: 969317
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Abstract: The problem is to determine when a smooth, $ k$-dimensional distribution $ {D^k}$ defined on an $ n$-manifold $ {M^n}$, locally admits a vector field basis which generates a nilpotent, solvable or even finite-dimensional Lie algebra. We show that for every $ 2 \leq k \leq n - 1$ there exists a (nonregular at $ p \in {M^n}$) distribution $ {D^k}$ on $ {M^n}$ which does not locally (near $ p$) admit a vector field basis generating a solvable Lie algebra. From classical results on the equivalence problem, it is shown that for $ 1 \leq k \leq 4$ and $ {D^k}$ regular at $ p \in {M^4}$, $ {D^k}$ admits a local vector field basis generating a nilpotent Lie algebra.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0969317-1
PII: S 0002-9939(1989)0969317-1
Article copyright: © Copyright 1989 American Mathematical Society