Double constructions of quadratic and sympletic antiassociative algebras

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.