Perturbations des superalgèbres de Lie. (Perturbations of Lie superalgebras) (Q922628)

Let \(L^ n_{p,q}\) be the algebraic variety of Lie superalgebra multiplications in \({\mathbb{C}}^ n={\mathbb{C}}^ p\oplus {\mathbb{C}}^ q\). In the general setting of nonstandard analysis, an element \(\mu\) of \(L^ n_{p,q}\) is said to be a perturbation of a standard \(\mu_ 0\in L^ n_{p,q}\) if \(\mu (X,Y)\simeq \mu_ 0(X,Y)\) for any standard X, Y in \({\mathbb{C}}^ n\). It is shown that, in this case \(\mu =\mu_ 0+\epsilon_ 1\phi_ 1+\epsilon_ 1\epsilon_ 2\phi_ 2+...+\epsilon_ 1...\epsilon_ k\phi_ k\), where the \(\epsilon_ i's\) are infinitesimal complex numbers and the \(\phi_ i's\) are linearly independent standard superanticommutative multiplications on \({\mathbb{C}}^ n={\mathbb{C}}^ p\oplus {\mathbb{C}}^ q\) satisfying several conditions. The standard multiplication \(\mu_ 0\in L^ n_{p,q}\) is said to be rigid in case any perturbation is isomorphic to it. Rigid Lie superalgebras are shown to exist for any n and p.