\(\delta\)-derivations of prime Lie algebras (Q1972200)

Let \(\Phi\) be an associative commutative ring with unity and let \(A\) be a \(\Phi\)-algebra. A \(\Phi\)-linear mapping \(\phi\: A \mapsto A \) satisfying the identity \((xy)\phi=\delta(x\phi)y+\delta x(y\phi)\) is called a \(\delta\)-derivation of~\(A\). In the first part of the article under review, the author proves the following theorems: 1. If \(\phi\) is a \(\delta\)-derivation of an arbitrary Lie \(\Phi\)-algebra \(A\) and if \(A\) is \(2\delta(\delta-1)(2\delta-1)\)-torsion free, then the fully characteristic ideal \(S_4(A)\) generated by all values of the standard Lie polynomial \(xs_4(y,z,t,v)\) lies in the kernel \(\text{Ker} \phi\) of \(\phi\). 2. Every prime Lie \(\Phi\)-algebra (\(1/2\in\Phi\)) has no nonzero \(\delta\)-derivations when \(\delta\neq -1,0,1/2,1\). In the second part, the author considers the case \(\delta=-1\) (in this event, the mapping \(\phi\) is called an antiderivation) and proves that every prime Lie \(\Phi\)-algebra \(A\) (\(1/6\in\Phi\)) with nonzero antiderivation satisfies the identity \([(yz)(tx)]x+[(yx)(zx)]t=0\) and is a \(3\)-dimensional central simple algebra over the fraction field of the center \(Z_R(A)\) of the algebra \(R(A)\) of right multiplications of \(A\). In the third part of the article, the author deals with the case \(\delta= 1/2\). Let \(\Delta(A)\) be the set of all \(1/2\)-derivations of \(A\) and let \(\Gamma(A)\) be the centroid of \(A\). In the previous article [Sib. Math. J. 39, No. 6, 1218-1230 (1998; Zbl 0936.17020)], the author proved that the equality \(\Delta(A)=\Gamma(A)\) holds for every prime Lie \(\Phi\)-algebra \(A\) \((1/6\in\Phi)\) equipped with a nondegenerate symmetric invariant bilinear form. In the general case, this assertion is false even for simple Lie algebras. The natural question arises of the description for the set \(\Delta(A)\) of a prime Lie algebra \(A\). A partial answer to this question is the author's assertion that the set \(\Delta(A)\) of an arbitrary prime Lie \(\Phi\)-algebra \(A\) \((1/6\in\Phi)\) is a commutative subalgebra of the endomorphism algebra \(\text{End}_{\Phi}(A)\) of the \(\Phi\)-module \(A\). From the author's conclusion: ``The algebra \(\Delta\) seems to play an essential role in describing various classes of prime (in particular, simple) Lie algebras''.