Classification of semisimple Levi-Tanaka algebras
A partial complex structure on a graded real Lie algebra \(g= \bigoplus_{p\in \mathbb{Z}} g_p\) is an \(\mathbb{R}\)-linear map \(J:g_{-1}\to g_{-1}\) such that \(J^2=- \text{Id}_{g_{-1}}\) and \([JX,JY]= [X,Y]\) \(\forall X,Y\in g_{-1}\). A Levi-Tanaka Lie algebra, which arises in the study of the group of pseudo-conformal automorphisms of a real submanifold \(M\) of a complex manifold \(X\) [cf. \textit{N. Tanaka}, J. Math. Kyoto Univ. 10, 1-82 (1970; Zbl 0206.50503)], is a graded real Lie algebra endowed with a partial complex structure satisfying certain (natural) conditions. In the article under review, the author gives a complete classification of the possible Levi-Tanaka algebra structures on real semisimple Lie algebras.
scientific article; zbMATH DE number 1463348
scientific article