A projective geometry of Lie algebras (Q957812)

Suppose that \(L\) and \(M\) are Lie algebras over rings \(R\) and \(S\) respectively. This paper finds cases when an isomorphism, \(f\), between the lattices of subalgebras of \(L\) and \(M\) implies that \(L\) and \(M\) are isomorphic and so are \(R\) and \(S\). An example is when \(R\) is a principal ideal ring and \(L\) is free of dimension greater than one. Another is when \(R\) and \(S\) are domains, \(R\) is a principal ideal ring and \(L\) is free polynilpotent. Other results show the existence of a semilinear isomorphism, \(u\), from \(L\) to \(M\) with respect to an isomorphism from \(R\) to \(S\) such that \(f(A)=u(A)\) for all subalgebras \(A\) of \(L\).