Sous-algèbres birégulières d'une algèbre de Kac-Moody-Borcherds

Abstract

Let ${\frak g}$ be a Kac-Moody-Borcherds algebra on a field ${\Bbb K}$ associated to a symetrizable matrix and with Cartan subalgebra ${\frak h}$. Let ${\frak L}$ be an ad$\,{\frak h}$-invariant subalgebra such that the restriction to ${\frak L}$ of the standard bilinear form is nondegenerate. We show that the root system $\Psi$ of $({\frak L}, {\frak h})$ is a subsystem according to [Ba] of $\Delta ({\frak g}, {\frak h})$. Moreover, if a subsystem $\Omega$ satisfies some conditions (i.e. $\Omega$ is "réduit et presque-clos") of $\Psi$, we construct inside of ${\frak L}$ a Kac-Moody-Borcherds algebra with root system $\Omega$.

Let $k$ be a subfield of ${\Bbb K}$. We prove similar results in the case of an action of a finite group of k-semi-automorphisms. In particular, we obtain a generalization to the Kac-Moody case of a result by Borel and Tits.

Let ${\frak g}$ be an almost-$k$-split form of a Kac-Moody algebra. We construct a Kac-Moody $k$-algebra with root system similar to the system of ${\frak g}$ (save on some multiples of certain roots).