Algebraic Results of Centralizers in Associative Algebras

Abstract In this document, we explore the algebraic properties of the centralizeror multiplier in different types of algebras, aiming to discern patternsand relationships. We introduce concepts such as Focal Algebra, NijenhuisOperator, and Reynolds Operator, and provide illustrative examples of theselinear applications within associative algebras defined by a left or right identityelement. Furthermore, we investigate the interaction between the centralizer andthe construction of specific algebraic structures within the framework of nonassociativealgebra theory. Essentially, these operators applied to certain associativealgebras give rise to non-associative algebraic structures, such as LieAlgebras or Dendriform Algebras. Our main achievement lies in the constructionof non-associative algebras, particularly dendriform algebras, derived fromassociative algebras in conjunction with a centralizer. For each construction,we provide concrete examples of non-associative algebras within subspaces ofsquare matrices.