Lie algebras associated with modules over polynomial rings

Authors

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

Abstract

Let 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 K. Define the Lie algebra LV=KP,QV as the semidirect product of two abelian Lie algebras with the natural action of KP,Q on V. We show that if K[x,y]-modules V and W are isomorphic or weakly isomorphic, then the corresponding associated Lie algebras LV and LW are isomorphic. The converse is not true: we 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 characterize such pairs of K[x,y]-modules of arbitrary dimension over K. We prove that indecomposable modules V and W with dimKV=dimKW7 are weakly isomorphic if and only if their associated Lie algebras LV and LW are isomorphic.

Published

25.09.2017

Issue

Section

Research articles

How to Cite

Petravchuk, A. P., and K. Ya. Sysak. “Lie Algebras Associated With Modules over Polynomial Rings”. Ukrains’kyi Matematychnyi Zhurnal, vol. 69, no. 9, Sept. 2017, pp. 1232-41, https://umj.imath.kiev.ua/index.php/umj/article/view/1773.