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)


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