Cohomology of oriented algebras

Abstract

We introduce a notion of oriented dialgebras and develop a cohomology theory for oriented dialgebras based on the possibility to mix the standard chain complexes computing group cohomology and associative dialgebra cohomologies. We also introduce a 1-parameter formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations.

Oriented Algebras and the Hochschild Cohomology Group

“…In [5], we proved thatH 1 G (A, X) classifies the singular extensions of oriented algebras. Here we obtain a similar result.…”

“…Definition 1. An orientation is a pair (G, ε) [5], where G is a group, and ε is a group homomorphism…”

“…Definition 2. Let G be an oriented group and A be an associative algebra [5]. An oriented action of (G, ε) on A is given by a map G × A → A,…”

“…Definition 3. Let A and B be oriented algebras over an oriented group (G, ε) [5]. A homomorphism of G-modules f : A → B is called a homomorphism of oriented algebras provided f is a homomorphism of algebras.…”

“…Definition 4. Let A be an oriented algebra over an oriented group (G, ε) [5]. An oriented bimodule over A is a usual bimodule X together with a G-module structure on X such that…”