Double constructions of quadratic and sympletic antiassociative algebras

TL;DR

This paper explores double constructions of quadratic and sympletic antiassociative algebras, providing classifications and detailed examples, especially for the case when q equals -1, highlighting their algebraic structures and properties.

Contribution

It introduces new double constructions of quadratic and sympletic antiassociative algebras, including classifications and explicit examples for the q=-1 case.

Findings

Constructed antiassociative algebras with specific decompositions

Classified 2-dimensional antiassociative algebras

Provided explicit double constructions and examples

Abstract

This work addresses some relevant characteristics and properties of $q$-generalized associative algebras and $q$-generalized dendriform algebras such as bimodules, matched pairs. We construct for the special case of $q=-1$ an antiassociative algebra with a decomposition into the direct sum of the underlying vector spaces of another antiassociative algebra and its dual such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is sympletic. The former is called a double construction of quadratic antiassociative algebra and the later is a double construction of sympletic antiassociative algebra which is interpreted in terms of antidendrifom algebras. We classify the 2-dimensional antiassociative algebras and thoroughly give some double constructions of quadratic and sympletic antiassociative algebras.