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Index and invariant polynomials of certain Lie algebras. (Indice et polynômes invariants pour certaines algèbres de Lie.) (French) Zbl 0743.17008
For a Lie algebra $${\mathfrak g}$$ and a positive integer $$m$$, we consider the Lie algebra $${\mathfrak g}_ m={\mathfrak g}+{\mathfrak g}\otimes T+\cdots+{\mathfrak g}\otimes T^ m$$ with bracket given by $$[x\otimes T^ r$$, $$y\otimes T^ s]=[x,y]\otimes T^{r+s}$$ if $$r+s\leq m$$, and 0 if $$r+s>m$$. In the first part of this work, we prove that the index of $${\mathfrak g}_ m$$ is $$(m+1)\chi({\mathfrak g})$$ (where $$\chi({\mathfrak g})$$ is the index of $${\mathfrak g})$$, and we show that $$(f_ 0,\ldots,f_ m)\in{\mathfrak g}^*_ m$$ (identified with $$({\mathfrak g}^*)^ m)$$ is regular if and only if $$f_ m$$ is a regular element in $${\mathfrak g}^*$$. In the second part, we suppose that $${\mathfrak g}$$ is semi-simple. We show that $$S({\mathfrak g}_ m)^{{\mathfrak g}_ m}$$ is a polynomial algebra in $$(m+1)r$$ indeterminates (where $$r$$ is the rank of $${\mathfrak g})$$, and we give a set of algebraically independent generators of this algebra. As in the semi- simple case, we construct a subspace $${\mathfrak t}_ m$$ of $${\mathfrak g}_ m$$ which has the following property: every regular adjoint orbit in $${\mathfrak g}_ m$$ meets $${\mathfrak t}_ m$$ in a single point, and this, in transversal manner.
Reviewer: M.Rais (Poitiers)

MSC:
 17B05 Structure theory for Lie algebras and superalgebras 17B20 Simple, semisimple, reductive (super)algebras
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