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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The formal linearization of a semisimple Lie algebra of vector fields about a singular point


Author: Robert Hermann
Journal: Trans. Amer. Math. Soc. 130 (1968), 105-109
MSC: Primary 22.90
MathSciNet review: 0217225
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Abstract: A classical theorem by Poincaré gives conditions that a nonlinear ordinary differential equation

$\displaystyle dx/dt = A(x),$

with $ A(0) = 0$ in n variables $ x = ({x_1}, \ldots ,{x_n})$ can be reduced to a linear form

$\displaystyle \frac{{dx'}}{{dt}} = \frac{{\partial A}}{{\partial x}}(0)x'$

by a change of variables $ x' = f(x)$. A generalization is given for a finite set of such differential equations, which form a semisimple Lie algebra.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1968-0217225-7
PII: S 0002-9947(1968)0217225-7
Article copyright: © Copyright 1968 American Mathematical Society