Affine Kac-Moody algebras graded by affine root systems
Résumé
The aim of this paper is to give the complete classification of all affine Kac-Moody algebras graded by affine root systems. An affine Lie algebra $gc$ is said graded by the affine root system $Si$ if $gc$ contains an affine subalgebra $ag$ whose root system relatively to a Cartan subalgebra $h_{ag}$ is equal to $Si$ and such that $gc=igopluslimits_{lambda in Si cup {0}} V_lambda$ where $V_lambda = { x in gc, [h,x]=lambda(h)x hs hbox{for all} hs h in h_{ag} }$. The Lie algebra $ag$ is then called the grading subalgebra. This is slightly modification of the notion of grading by a finite root system which was introduced by Bermann and Moody ([BM]) and developed by Benkart-Zelmanov ([BZ]), Neher ([Ne]), Allison-Benkart-Gao ([ABG]). In a preceding paper ([N]) we gave the complete classification of all simple finite dimensional Lie algebras graded by a finite root system.
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