On a problem of V. G. Kac: the classification of certain simple graded Lie algebras
The author proves the following theorem: Let \(K\) be a field of characteristic zero. Let \(L=\oplus L_n\) be a Lie algebra over \(K\), graded over \(\mathbb{Z}\) such that (1) \(L\) is simple graded and (2) \(\dim L_n\leq 1\) \(\forall n\in \mathbb{Z}\). Then \(L\) is isomorphic to one of the following Lie algebras suitably graded: \[ \mathfrak{sl}(2), \quad A_1^{(1)}, \quad A_2^{(2)}, \quad W_1, \quad W. \] Here \(W\) is the Witt algebra, i.e. it has a basis \(\{u_i\}_{i\in\mathbb{Z}}\) such that \([u_i,u_j] = (j -i)u_{i+j}\) and \(W_1\) is the subalgebra of \(W\) with basis \(\{u_i\}_{i\ge -1}\). The case when \(\dim L_n=1\) \(\forall n\) had been conjectured by V. G. Kac. The following lemma was very useful in the demonstration: Let \(L\) be a \(\mathbb{Z}\)-graded Lie algebra and let \(A\) and \(B\) be two graded vector subspaces such that \(L=A+B\) and \([A,B]\subset B\). Then \(B+[B,B]\) is a graded ideal. [This article is identical to the author's thesis, Univ. Paris VII, 57 p. (1984).]
scientific article; zbMATH DE number 3959724
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