广义 prelie 代数与置换代数
Generalized prelie and permutative algebras
AI总结：
本文引入由双线性映射 $Φ$ 控制的边类型有根树与广义嫁接积，构造 PreLie $Φ$ operad、余相互作用双代数对及其 Koszul 对偶，统一推广 pre-Lie 与置换代数。
摘要：
我们研究 pre-Lie 代数的推广，其中自由对象建立在边带有类型的有根树之上，而非通常的有根树，并且广义 pre-Lie 积由嫁接构成。在使用离散类型集合时，我们说明当该集合被赋予一个结合交换积以及使其成为交换扩展半群的第二个积时，如何获得此类对象。在使用类型向量空间时，这两个积被一个满足辫子方程和交换关系的双线性映射 $Φ$ 所替代。此类结构的例子定义在集合、半群或群上。这些构造定义了一族 operad PreLie $Φ$，它们推广了 pre-Lie 代数的 operad PreLie。对于从 PreLie 到 PreLie $ϕ$ 的任意嵌入，我们基于带类型和装饰的树构造了一族余相互作用双代数对：第一个余积由提取与收缩过程给出，其中类型通过 $Φ$ 的作用被修改；第二个余积以 Connes 和 Kreimer 的方式由容许切割给出，类型同样通过 $Φ$ 的作用被修改。我们还研究了 PreLie $Φ$ 的 Koszul 对偶，由此得到置换代数的推广。
英文摘要：
We study generalizations of pre-Lie algebras, where the free objects are based on rooted trees which edges are typed, instead of usual rooted trees, and with generalized pre-Lie products formed by graftings. Working with a discrete set of types, we show how to obtain such objects when this set is given an associative commutative product and a second product making it a commutative extended semigroup. Working with a vector space of types, these two products are replaced by a bilinear map $Φ$ which satisfies a braid equation and a commutation relation. Examples of such structures are defined on sets, semigroups, or groups. These constructions define a family of operads PreLie $Φ$ which generalize the operad of pre-Lie algebras PreLie. For any embedding from PreLie into PreLie $ϕ$ , we construct a family of pairs of cointeracting bialgebras, based on typed and decorated trees: the first coproduct is given by an extraction and contraction process, the types being modified by the action of $Φ$; the second coproduct is given by admissible cuts, in the Connes and Kreimer's way, with again types modified by the action of $Φ$. We also study the Koszul dual of PreLie $Φ$ , which gives generalizations of permutative algebras.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
