A Generalization of Operads Based on Subgraph Contractions

Pub Date : 2024-05-23 DOI:10.1093/imrn/rnae096
Denis Lyskov
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Abstract

We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $\operatorname{\overline{{\mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröbner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.
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基于子图收缩的 Operads 广义
我们引入了一种对操作数概念的概括,称之为契约数,其操作集以连通图为索引,其组成规则以连通子图的契约为编号。我们证明,许多经典的操作数,如交换代数、李代数、关联代数、前李代数的操作数,小磁盘操作数,以及稳定曲线模空间的操作数 $\operatorname{\overline{\mathcal{M}}}}_{0,n+1}$ 都可以概括为契约数。我们解释说,科斯祖尔对偶性和格洛布纳基机制等标准工具可以很容易地推广到 contractads。我们验证了交换、Lie、联立和格氏约元的科斯祖尔性质。
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