Anti-commutative anti-associative algebras. Acaa-algebras
Elisabeth Remm
TL;DR
The paper introduces Acaa algebras, defined by an anticommutative product that satisfies a quadratic identity yielding antiassociativity, and analyzes their structure via polarization. It establishes a characterization linking Acaa to a simple condition $[x_1,[x_2,x_1]]=0$, provides a complete classification in low dimensions, and describes free Acaa algebras. It then develops representation theory through the adjoint action and admissible algebras, and finally develops a cohomology theory and quadratic operad framework showing that the Acaa operad is not Koszul. The results illuminate the algebraic landscape of anticommutative anti-associative structures and connect them to familiar objects like Lie algebras in low dimensions, while exposing nontrivial operadic properties and cohomology for further study.
Abstract
Let $(A,μ)$ be a nonassociative algebra over a field of characteristic zero. The polarization process allows us to associate two other algebras, and this correspondence is one-one, one commutative, the other anti-commutative. Assume that $μ$ satisfies a quadratic identity $\sum_{σ\in Σ_3} a_σμ(μ(x_{σ(i)},x_{σ(j)}),x_{σ(k)}-a_σμ(x_{σ(i)},μ(x_{σ(j)},x_{σ(k)})=0.$ Under certain conditions, the polarization of such a multiplication determines an anticommutative multiplication also verifying a quadratic identity. Now only two identities are possible, the first is the Jacobi identity which makes this anticommutative multiplication a Lie algebra and the multiplication $μ$ is Lie admissible, the second, less classical is given by $[[x,y],z]=[[y,z],x]=[[z,x],y].$ Such a multiplication is here called Acaa for Anticommutative and Antiassociative. We establish some properties of this type of algebras.