二次与辛反结合代数的双重构造
Double constructions of quadratic and sympletic antiassociative algebras
摘要：
本文研究了$q$ -广义结合代数和$q$ -广义树状代数的一些相关特征和性质，如双模、匹配对等.对于$q=-1$的特殊情况，我们构造了一个反结合代数，它分解为另一个反结合代数的基础向量空间及其对偶的直和，使得它们都是子代数，并且自然的对称双线性型是不变的或自然的反对称双线性型是辛的.前者称为二次反结合代数的二重构造，后者称为辛反结合代数的二重构造，并用反枝形代数来解释。对二维反结合代数进行了分类，并给出了二次反结合代数和辛反结合代数的一些二重结构。
英文摘要：
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.
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