Lie algebras associated with modules over polynomial rings

  • A. P. Petravchuk Kyiv Nat. Taras Shevchenko Univ., Ukraine
  • K. Ya. Sysak

Abstract

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.
Published
25.09.2017
How to Cite
PetravchukA. P., and SysakK. Y. “Lie Algebras Associated With Modules over Polynomial Rings”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, no. 9, Sept. 2017, pp. 1232-41, http://umj.imath.kiev.ua/index.php/umj/article/view/1773.
Section
Research articles