Non-additive Lie centralizer of infinite strictly upper triangular matrices

Subject Areas : Linear and multilinear algebra; matrix theory

D. A. Aiat ‎Hadj ¹

1 - Department of Mathematics, Centre R\'{e}gional des M\'{e}tiers d'Education et de Formation (CRMEF) Tangier, Morocco

Received: 2019-01-22 Accepted : 2019-11-08 Published : 2019-12-01

Keywords: Lie centralizer, strictly upper triangular matrices, commuting map,

Abstract :

‎Let $\mathcal{F}$ be an field of zero characteristic and $N_{\infty‎}(‎\mathcal{F})$ be the algebra of infinite strictly upper triangular‎‎matrices with entries in $\mathcal{F}$‎, ‎and $f:N_{\infty}(\mathcal{F}‎)\rightarrow N_{\infty}(\mathcal{F})$ be a non-additive Lie centralizer of $‎N_{\infty }(\mathcal{F})$; that is‎, ‎a map satisfying that $f([X,Y])=[f(X),Y]$‎‎for all $X,Y\in N_{\infty}(\mathcal{F})$‎. ‎We prove that $f(X)=\lambda X$‎,‎where $\lambda \in \mathcal{F}$‎.

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