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Journal of Vibration Testing and System Dynamics 7(1) (2023) 5--13 | DOI:10.5890/JVTSD.2023.03.002

R. Campoamor-Stursberg

Instituto de Matem'atica Interdisciplinar IMI-UCM, Plaza de Ciencias 3, E-28040 Madrid, Spain

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Abstract

Vector field realizations of Lie algebras in connection with the representation theory and the branching rule problem associated to embeddings of semisimple Lie algebras are considered. This allows to determine if a subalgebra in a given realization corresponds to an irreducible embedding, as well as to determine multiplicities in the branching rules. The invariants of the realizations associated to such embeddings are used to construct (second-order) systems of ordinary differential equations possessing a fixed Lie algebra of point symmetries. It is shown that for any embedding $\mathfrak{g}^{\prime}\subset\mathfrak{g}$ and any faithful representation there exists an integer $k$ such that for any $n\geq k$, systems of order $n$ with exact Lie point symmetry $\mathfrak{g}^{\prime}$ can be constructed.

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