Inclusions of varieties generated by simple Lie algebras (Q919445)

It was proved earlier by the author and A. H. Kushkulej that any prime (simple) finite-dimensional algebra over an algebraically closed field K is defined by its identities to within isomorphism. Now the author takes an interest in the following question: is it true that the inclusion Var \({\mathfrak g}_ 1\subset Var {\mathfrak g}_ 2\) for simple finite-dimensional Lie algebras \({\mathfrak g}_ 1\) and \({\mathfrak g}_ 2\) implies the inclusion \({\mathfrak g}_ 1\subset {\mathfrak g}_ 2?\) The negative answer to this question for the modular case gives the following Theorem. Let char K\(=p\geq 5\), \({\mathfrak g}_ 1\) be a simple Hamiltonian Lie algebra over K of Cartan type \(H_ n\) and \({\mathfrak g}_ 2\) be a classical simple Lie K-algebra of type \(A_{p^ n-1}\). Then \[ Var {\mathfrak g}_ 1\subset Var {\mathfrak g}_ 2,\quad \dim_ K{\mathfrak g}_ 1=\dim_ K{\mathfrak g}_ 2=p^{2n}-2, \] but the algebras \({\mathfrak g}_ 1\) and \({\mathfrak g}_ 2\) are not isomorphic.