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Let  be an algebraically closed field with characteristic zero and let  be a module over the polynomial ring [] The actions of  and  specify linear operators  and  on  regarded as a vector space over  We define the Lie algebra  = 〈, 〉  as the semidirect product of two Abelian Lie algebras with the natural action of 〈, 〉 on  It is shown that if []-modules  and  are isomorphic or weakly isomorphic, then the corresponding associated Lie algebras  and  are isomorphic. The converse assertion is not true: we can construct two []-modules  and  of dimension 4 that are not weakly isomorphic but their associated Lie algebras are isomorphic. We present the characterization of these pairs of []-modules of arbitrary dimension over  It is shown that the indecomposable modules  and  with dim  = dim  7 are weakly isomorphic if and only if their associated Lie algebras  and  are isomorphic.

Lie Algebras Associated with Modules over Polynomial Rings