03-237 Rutwig Campoamor-Stursberg

The structure of the invariants of perfect Lie algebras (407K, Postscript) May 23, 03

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Abstract. Upper bounds for the number $\mathcal{N}(\frak{g})$ of Casimir operators of perfect Lie algebras $\frak{g}$ with nontrivial Levi decomposition are obtained, and in particular the existence of nontrivial invariants is proved. It is shown that for high ranked representations $R$ the Casimir operators of the semidirect sum $\frak{s}\overrightarrow{\oplus}_{R}(\deg R)L_{1}$ of a semisimple Lie algebra $\frak{s}$ and an abelian Lie algebra $(\deg R)L_{1}$ of dimension equal to the degree of $R$ are completely determined by the representation $R$, which also allows the analysis of the invariants of subalgebras which extend to operators of the total algebra. In particular, for the adjoint representation of a semisimple Lie algebra the Casimir operators of $\frak{s}\overrightarrow{\oplus}_{ad(\frak{s})}(\dim \frak{s})L_{1}$ can explicitely be constructed from the Casimir operators of the Levi part $\frak{s}$.

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