Lie Algebras Associated with Modules over Polynomial Rings

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