Steenrod operator of the dihedral homology of A_∞-algebras

In this study, we examine the interaction between Steenrod operations and dihedral homology within the framework of A-infinity algebras, aiming to understand how these operations, initial in algebraic topology, contribute to homological invariants and influence dihedral homology structures. We begin by exploring the initial properties of A-infinity algebras as generalizations of associative algebras, focusing on their volume to support Steenrod operations. From there, we delve into the effects of these operations within dihedral homology, uncovering their role in revealing deeper algebraic structures and improving our understanding of homology theories. We also analyze the connections between Steenrod operations and projective varieties over finite fields, emphasizing their actions in derived categories and their significance in the context of α-characteristic fields. By defining Steenrod operators within dihedral homology, we explain the complex relationships between these algebraic structures and operations derived from homology theory. Through specific examples and theoretical models, we demonstrate how these interactions advance our understanding of homological invariants and provide valuable tools and perspectives for the broader fields of algebraic topology and homological algebra.