Anti-quadratic nearly associative algebras

Abstract

An anti-quadratic nearly associative algebra $(A,\bullet ,\omega )$ is a nearly associative algebra equipped with a non-degenerate skew-symmetric bilinear form $\omega$ satisfying the anti-associativity condition $\omega (u\bullet v,w)=-\omega (u,v\bullet w)$ for all $u,v,w\in A.$ In this paper, we investigate the structural properties and extension theory of these algebras. We prove that any anti-quadratic nearly associative algebra is inherently nilpotent of index at most $6$. Furthermore, we introduce a novel $T{T}^{\ast }$-extension method and show that every anti-quadratic nearly associative algebra arises from this construction. We also introduce the notion of a double extension and prove that every such algebra can be constructed by iterated double extensions. Finally, as an application, we give a complete classification of anti-quadratic nearly associative algebras of dimension at most $4$.

arXiv comment24 pages