Associative-Yamaguti algebras

Abstract

In this paper, we first introduce associative-Yamaguti algebras as the associative analogue of Lie-Yamaguti algebras. Associative algebras, reductive associative algebras and associative triple systems of the first kind form subclasses of associative-Yamaguti algebras. Any diassociative algebra canonically provides an associative-Yamaguti algebra structure. We confirm that any associative-Yamaguti algebra admits an enveloping associative algebra (i.e., it can be obtained from a reductive associative algebra). We show that a suitable skew-symmetrization of an associative-Yamaguti algebra gives rise to a Lie-Yamaguti algebra structure. Next, we define the $(2,3)$-cohomology group of an associative-Yamaguti algebra to study formal one-parameter deformations and abelian extensions. Later, we consider Yamaguti multiplications on a nonsymmetric operad as a generalization of associative-Yamaguti algebras. This notion further leads us to introduce dendriform-Yamaguti algebras, which are splitting objects for associative-Yamaguti algebras. Finally, we consider relative Rota-Baxter operators on associative-Yamaguti algebras to establish close relationships with dendriform-Yamaguti algebras.