On Lie algebras associated with modules over polynomial rings

Let $\mathbb K$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $\mathbb K[x,y]$. The actions of $x$ and $y$ determine linear operators $P$ and $Q$ on $V$ as a vector space over $\mathbb K$. Define the Lie algebra $L_V=\mathbb K\langle P,Q\rangle \rightthreetimes V$ as the semidirect product of two abelian Lie algebras with the natural action of $\mathbb K\langle P,Q\rangle$ on $V$. We show that if $\mathbb K[x,y]$-modules $V$ and $W$ are isomorphic or weakly isomorphic, then the corresponding associated Lie algebras $L_V$ and $L_W$ are isomorphic. The converse is not true: we construct two $\mathbb K[x,y]$-modules $V$ and $W$ of dimension $4$ that are not weakly isomorphic but their associated Lie algebras are isomorphic. We characterize such pairs of $\mathbb K[x, y]$-modules of arbitrary dimension. We prove that indecomposable modules $V$ and $W$ with $\dim V=\dim W\geq 7$ are weakly isomorphic if and only if their associated Lie algebras $L_V$ and $L_W$ are isomorphic.