On \(\delta\)-derivations of prime Lie algebras
type: Item
instance of: scholarly article
title: On \(\delta\)-derivations of prime Lie algebras
zbMATH Open document ID: 0977.17015
author: V. T. Filippov
published in: Doklady Mathematics
publication date: 2001-01-28T00:00:00Z
review text: Let \(k\) be a commutative associative ring with unit element and with a fixed element \(\delta\). A \(\delta\)-derivation of a nonassociative \(k\)-algebra \(A\) is a \(k\)-module endomorphism \(\partial\) of \(A\) such that \(\partial(xy)=\delta(\partial x)y+\delta x(\partial y)\). If \(A\) is a Lie \(k\)-algebra and \(\delta(\delta-1)\) is invertible in \(k\), then \(A\) with a new multiplication \(x\cdot y=\partial(xy)\) is again a Lie \(k\)-algebra. Suppose that \(2\delta(\delta-1)(2\delta-1)\) is invertible in \(k\). Then the ideal generated by all evaluations of the standard polynomial \(xS_4(y,z,t,v)\) is annihilated by a \(\delta\)-derivation. Any prime Lie \(k\)-algebra has no non-zero \(\delta\)-derivation if 2 is invertible in \(k\) and \(\delta\neq \pm 1, 0,\frac 12\). If a prime Lie \(k\)-algebra \(A\) has no \((-1)\)-derivations and 6 is invertible in \(k\), then \[ \left[\left[ [y,z],[t,x]\right],x\right]+ \left[\left[ [y,x],[z,x]\right],t\right]=0 \] for all \(x,y,z,t\in A\). If a prime Lie \(k\)-algebra \(A\) has a non-zero \((-1)\)-derivation and 6 is invertible in \(k\), then \(A\) is a three-dimensional central simple algebra. There are some other results of the similar nature.
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 1557603
zbMATH Keywords: derivations
zbMATH Keywords: prime Lie algebras
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1557603