正则参数化单关系运算的分类
Classification of regular parametrized one-relation operads
摘要：
Jean-Louis Loday引入了一类对称运算，它由一个双线性运算生成，服从一个关系，使得三个元素的每个左赋范乘积等于右赋范乘积的线性组合：\[ (a_1a_2)a_3=\sum_{σ\in S_3}x_σ\, a_{σ(1)}(a_{σ(2)}a_{σ(3)})\ ; \]这样的运算称为参数化单关系运算。对于参数$\{x_σ\}$的特定选择，如果它的每一个分量都是对称群的正则表示，则称这个运算是正则的;等价地，向量空间$V$上相应的自由代数作为分次向量空间，同构于$V$的张量代数。在特征为零的代数闭域上，我们对所有正则参数化的单关系运算进行了分类。事实上，我们证明，每个这样的操作同构于以下五个操作之一：左幂零操作定义的身份$((a_1a_2)a_3)=0$，结合操作，莱布尼茨操作，对偶莱布尼茨（Zinbiel）操作，和泊松操作。我们的计算方法结合线性代数多项式环，对称群的表示论，和Gröbner基地的行列式理想和他们的根。
英文摘要：
Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: \[ (a_1a_2)a_3=\sum_{σ\in S_3}x_σ\, a_{σ(1)}(a_{σ(2)}a_{σ(3)})\ ; \] such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{x_σ\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the identity $((a_1a_2)a_3)=0$, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.
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