On Lie algebras associated with modules over polynomial rings

TL;DR

This paper explores the relationship between modules over polynomial rings and their associated Lie algebras, establishing conditions under which module isomorphisms correspond to Lie algebra isomorphisms and identifying exceptions.

Contribution

It introduces a construction linking polynomial ring modules to Lie algebras and characterizes when these Lie algebras reflect module isomorphisms, including new counterexamples.

Findings

Isomorphic modules lead to isomorphic Lie algebras.

Counterexamples show non-isomorphic modules can have isomorphic Lie algebras.

Indecomposable modules of dimension ≥7 are weakly isomorphic iff their Lie algebras are isomorphic.

Abstract

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.