Characterizing Linear Maps Preserving Lie Products and Operator Products

This paper investigates the properties of linear maps that preserve specific algebraic structures, namely Lie products (commutators) and operator products (anti-commutators). The core contribution lies in characterizing the general form of these maps under the constraint that the product of the input elements maps to a fixed element. This is relevant to understanding structure-preserving transformations in linear algebra and operator theory, potentially impacting areas like quantum mechanics and operator algebras. The paper's significance lies in providing a complete characterization of these maps, which can be used to understand the behavior of these products under transformations.