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Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators. (English) Zbl 1156.17003
The authors study the universal enveloping algebra $U(\mathfrak{g})$ of the semidirect product $\mathfrak{g}$ of a semisimple Lie algebra $\mathfrak{s}$ with a solvable Lie algebra $\mathfrak{r}$. The authors determine certain elements in $U(\mathfrak{g})$ which commute with $\mathfrak{r}$. It turns out that, up to a functorial factor given by a Casimir operator of $\mathfrak{g}$, the commutators or these elements are similar to the commutators of generators of the algebra $\mathfrak{s}$. These elements generate what is called a virtual copy of $\mathfrak{s}$ in $U(\mathfrak{g})$ (as the normalizer version of these elements formally belongs only to the (skew)-field of fractions of $U(\mathfrak{g})$ and not to $U(\mathfrak{g})$ itself). Using this virtual copy the authors compute Casimir elements in $U(\mathfrak{g})$. As an application the authors obtain a formula for Casimir operators for the inhomogeneous Hamilton algebras $\mathrm{IHa}(N)$ and all its central extensions in arbitrary dimensions. The authors also study the behavior of virtual copies with respect to contractions.

##### MSC:
 17B35 Universal enveloping Lie (super)algebras 17B05 Structure theory of Lie algebras
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