Notes on the Lie algebra \(\Sigma (n,m,{\mathfrak r},G)\) (Q1096710)

The author [ibid. 4, 328-346 (1986; Zbl 0507.17007)] constructed explicitly a class of Lie algebras \(\Sigma =\Sigma (n,m,{\mathfrak r},G)\) over a field F of characteristic \(p>2\), from which three classes of simple Lie algebras \({\bar \Sigma}\), \(\Sigma\) * and \({\tilde \Sigma}\) are constructed and \(\Sigma\) * and \({\tilde \Sigma}\) are shown to be new (i.e., not included in the list of ``known simple Lie algebras'' of \textit{R. L. Wilson} [J. Algebra 40, 418-465 (1976; Zbl 0355.17012)]). This note gives supplementary facts about \(\Sigma\). (1) \({\bar \Sigma}\) is isomorphic to a simple Lie algebra associated with a nodal noncommutative Jordan algebra. (2) If Char F\(=2\), \(\Sigma\) gives simple Lie algebras as well. Especially, a class of simple Lie algebras of Cartan type K is obtained (note that the usual normalized algebra of Cartan type K is not simple). (3) If Char F\(=0\), \(\Sigma\) is a simple infinite dimensional graded Lie algebra whose zero grade term is, roughly, a loop algebra \(P\otimes {\mathfrak sp}(2n)\), where P is the Laurent polynomial algebra, extended by the differential operator on P. The note also includes corrections of some minor errors of the author's paper (loc. cit.).