Lie algebras associated with modules over polynomial rings

• Datum: 2017
• Hauptverfasser: Petravchuk, A. P., Sysak, K. Ya., Петравчук, А. П., Сисак, К. Я.
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2017
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/1773
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $K[x, y]$. 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 $L_V = K\langle P,Q\rangle \rightthreetimes V$ 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 over K. We prove that indecomposable modules $V$ and $W$ with $\mathrm{d}\mathrm{i}\mathrm{m}\mathbb{K} V = \mathrm{d}\mathrm{i}\mathrm{m}KW \geq 7$ are weakly isomorphic if and only if their associated Lie algebras $L_V$ and $L_W$ are isomorphic.