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On the index and biderivations of simple Malcev algebras

Abdelaziz Ben Yahya|Said Boulmane
대한수학회 (The Korean Mathematical Society)|2022.04|대한수학회논문집 (Communications of the KMS)|제37권 제2호|pp. 385-397|DOI 10.4134/CKMS.c210156

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키워드

Lie algebrasMalcev algebrasskew-symmetric biderivationsLie triple systemsquadratic Malcev algebra.
주제분류수학

초록

Let $(M,[\;,\;])$ be a finite dimensional Malcev algebra over an algebraically closed field $\mathbb{F}$ of characteristic 0. We first prove that, $(M,[\;,\;])$ (with $[M,M]\neq 0$) is simple if and only if $ind(M)=1$ (i.e., $M$ admits a unique (up to a scalar multiple) invariant scalar product). Further, we characterize the form of skew-symmetric biderivations on simple Malcev algebras. In particular, we prove that the simple seven dimensional non-Lie Malcev algebra has no nontrivial skew-symmetric biderivation.

Abstract

Let $(M,[\;,\;])$ be a finite dimensional Malcev algebra over an algebraically closed field $\mathbb{F}$ of characteristic 0. We first prove that, $(M,[\;,\;])$ (with $[M,M]\neq 0$) is simple if and only if $ind(M)=1$ (i.e., $M$ admits a unique (up to a scalar multiple) invariant scalar product). Further, we characterize the form of skew-symmetric biderivations on simple Malcev algebras. In particular, we prove that the simple seven dimensional non-Lie Malcev algebra has no nontrivial skew-symmetric biderivation.

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