Algebras satisfying Capelli identities

In this paper the author considers the representation of an algebra $L$ of a certain signature in an algebra $A$ (generally of a different signature) satisfying identity relations of Capelli type. A criterion for the Capelli identities to hold in the pair $(A,L)$ is indicated, and a structural description of such pairs is given. The results are applied for the case where $L$ is a Lie algebra and $A$ is its associative enveloping algebra. In addition, from these results it is deduced that an “algebraicity” identity over a field of characteristic zero implies nilpotence of the Jacobson radical of a finitely generated associative algebra.
Bibliography: 10 titles.