Quasiderivations and quasicentroids of Novikov algebras
A Novikov algebra is a nonassociative algebra satisfying the identities \(x(yz)-y(xz)=(xy-yx)z\) (left-symmetric algebra) and \((xy)z=(xz)y\). Given a nonassociative algebra \(A\), a linear endomorphism \(D\) of \(A\) is said to be a quasiderivation if there is a linear endomorphism \(D'\) such that \(D(x)y+xD(y)=D'(xy)\) for all \(x,y\in A\). \par The main result of the paper under review asserts that if \(A\) is a finite-dimensional Novikov algebra over a field of characteristic not two with \(A^2\neq 0\), such that any linear endomorphism of \(A\) is a quasiderivation, then the dimension of \(A\) is at most \(3\). Moreover, for any Novikov algebra \(A\), a nilpotent algebra \(B\), with \(B^3=0\), is constructed such that the Lie algebra of quasiderivations of \(A\) is naturally embedded in the Lie algebra of derivations of \(B\). The algebra \(B\) is the tensor product of \(A\) with the unique nontrivial nilpotent commutative associative algebra of dimension \(2\). Remark: Some of the results, and their proofs, are valid for arbitrary nonassociative algebras.
scientific article; zbMATH DE number 7377338
scientific article