Symmetric $(\sigma ,\tau )$-algebras and $(\sigma ,\tau )$-Hochschild cohomology

Abstract

On an associative algebra, we introduce the concept of symmetric $(\sigma ,\tau )$-derivations together with a regularity condition and prove that strongly regular symmetric $(\sigma ,\tau )$-derivations are inner. Symmetric $(\sigma ,\tau )$-derivations are $(\sigma ,\tau )$-derivations that are simultaneously $(\sigma ,\tau )$-derivations as well as $(\tau ,\sigma )$-derivations, generalizing a property of commutative algebras. Motivated by this notion, we explore the geometry of symmetric $(\sigma ,\tau )$-algebras and prove that there exist a unique strongly regular symmetric $(\sigma ,\tau )$-connection. Furthermore, we introduce $(\sigma ,\tau )$-Hochschild cohomology and show that, in first degree, it describes the outer $(\sigma ,\tau )$-derivations on an associative algebra. Along the way, examples are provided to illustrate the novel concepts.

25 pages