On Lie algebras associated with modules over polynomial rings

2017/01/13 by А. П. Петравчук, Petravchuk, A. P., K. Ya. Sysak +1
Mathematics · #Advanced Topics in Algebra #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1701.03750

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⟨ P,Q⟩ \rightthreetimes V as the semidirect product of two abelian Lie algebras with the natural action of \mathbb K⟨ P,Q⟩ 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≥ 7 are weakly isomorphic if and only if their associated Lie algebras L_V and L_W are isomorphic.

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