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On an associative algebra, we introduce the concept of symmetric$(\sigma,\tau)$-derivations together with a regularity condition and prove thatstrongly regular symmetric $(\sigma,\tau)$-derivations are inner. Symmetric$(\sigma,\tau)$-derivations are $(\sigma,\tau)$-derivations that aresimultaneously $(\sigma,\tau)$-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 regularsymmetric $(\sigma,\tau)$-connection. Furthermore, we introduce$(\sigma,\tau)$-Hochschild cohomology and show that, in first degree, itdescribes the outer $(\sigma,\tau)$-derivations on an associative algebra.Along the way, examples are provided to illustrate the novel concepts.

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