On the structure of a Lie-admissible algebra in the space of Gâteaux differentiable operators
By an algebra we shall mean a linear, in general, nonassociative algebra over the field \(\mathbb{R}\) of real numbers. An algebra \(\langle A,(*)\rangle\) is said to be Lie-admissible if the linear space \(A\), endowed by the bilinear operation \[ [a, b]= a* b- b* a\qquad \forall a, b\in A \] is a Lie algebra. Structures of the mentioned form have applications in classical mechanics and in theoretical physics. In connection with this it is of interest to describe various Lie-admissible operations \((*)\) in given linear spaces. The purpose of this note is to describe a very general structure of a Lie-admissible algebra in the real linear space of Gâteaux differentiable operators.
real linear space of Gâteaux differentiable operators
scientific article; zbMATH DE number 796775