Index and invariant polynomials of certain Lie algebras. (Indice et polynômes invariants pour certaines algèbres de Lie.) (French) Zbl 0743.17008

Rais, Mustapha; Tauvel, Patrice

J. Reine Angew. Math. 425, 123-140 (1992).

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

Keywords:

index; regular element; semi-simple; polynomial algebra; generators