On the Dihedral Homology of A∞-algebras

In this paper, we will show and discuss the Steenrod operators and their applications within the framework of the dihedral homology of A-infinity algebras. Steenrod operations have proved to be important tools in developing the study of homological elements and various homological theories, such as the Adams spectral sequence and the Sullivan conjecture, as they were first introduced in algebraic topology. These operators have proven to be invaluable in advancing our understanding of topological and algebraic structures. Therefore, we focus on the generalization of these methods for more general applications, particularly with projective homogeneous varieties over α-characteristic fields. We begin by defining Steenrod operators in dihedral homology of A-infinity algebras, and we will explain the complex relations between these algebraic structures and the homology theory-derived operations.