LAUSR

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