Hom-structures and bihom-structures of antiflexible and pre-antiflexible algebras

Abstract

UDC 512.554, 512.628

We define and study the structures of Pre-Anti-Flexible algebras in the Hom- and BiHom-cases. More precisely, in the BiHom-case, the corresponding algebraic structure is defined by two products $\triangleleft,$ $\triangleright$ and two linear maps $f$ and $g$ on $A.$ This structure is denoted by $\big(A,\triangleleft,\triangleright,f,g\big)$ and if $f=g,$ then we get the Hom-version denoted by $\big(A,\triangleleft,\triangleright,f\big).$ Our main results can be desribed as follows:  we define some algebraic structures and investigate some properties and relationships between the BiHom-Pre-Anti-Flexible algebras and BiHom-Anti-Flexible algebras. Furthermore, we prove that any BiHom-Anti-Flexible algebra equipped with a Rota–Baxter operator defines a BiHom-Pre-Anti-Flexible algebra. Finally, we introduce the notion of Nijenhuis operator in the BiHom setting and demonstrate that a Nijenhuis operator on a BiHom-Anti-Flexible algebra specifies a BiHom-Pre-Anti-Flexible algebra.