On the construction of complete Lie algebras (Q1902149)

This paper describes a method for the construction of complete Lie algebras over a field of characteristic zero. Notions from the fields of algebraic Lie algebras and algebraic groups are used. Let \(S\) be a semisimple Lie algebra acting on the nilpotent Lie algebra \(N\). Let \(D\) be the centralizer of the image of \(S\) in \(\text{Der} (N)\) and let \(C\) be a Cartan subalgebra of \(D\) with \(T\) being the subalgebra of semisimple elements of \(C\). Let \(G = (S \oplus C) + N\), \(M = S \oplus T\) and \(A\) be the centralizer of \(M + N\) in \(G\). Then \(G/A\) is complete along with another condition if and only if the centralizer of \(Z(N)\) in \(M\) is 0 and \(N = [M,N]\). Furthermore, all complete Lie algebras can be constructed this way.