Simple Sagle algebras
type: Item
instance of: scholarly article
title: Simple Sagle algebras
zbMATH Open document ID: 0969.17001
DOI: 10.1007/BF02675366
author: V. T. Filippov
published in: Mathematical Notes
publication date: 2001-09-27T00:00:00Z
review text: A Sagle algebra over a field is a nonassociative algebra satisfying the identities \[ x^2=0,\quad J(x,y,z)w=J(w,z,xy)+J(w,y,zx)+J(w,x,yz) \] where \(J(x,y,z)=(xy)z+(yz)x+(zx)y\). Each Sagle algebra is a 3-Lie algebra in the sense of the author with respect to the triple multiplication \(\langle x,y,z\rangle =J(x,y,z)\). It is shown that if the characteristic of the base field is not equal to 3, then a non-Lie Sagle algebra is simple if and only if is it simple as a 3-Lie algebra. An arbitrary finite-dimensional simple Sagle algebra over a field of characteristic zero is a Lie algebra. The last result solves a problem raised by \textit{A. A. Sagle} in Pac. J. Math. 15, 1397-1400 (1965; Zbl 0241.17007).
Mathematics Subject Classification ID: 17A40
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 1470958
zbMATH Keywords: 3-Lie algebras
zbMATH Keywords: triple systems
zbMATH Keywords: Sagle algebra
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
cites work: Q3252368
cites work: On anti-commutative algebras and general Lie triple systems
cites work: Q4011919
cites work: On simple algebras obtained from homogeneous general Lie triple systems
cites work: Q4302405
cites work: Q3710656
description: scientific article; zbMATH DE number 1470958