On \(\delta\)-derivations of prime Lie algebras
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.
scientific article; zbMATH DE number 1557603