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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.

References [Enhancements On Off] (What's this?)

  • [1] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
  • [2] Shlomo Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824. MR 0096853

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1968-0217225-7
Article copyright: © Copyright 1968 American Mathematical Society