Operads from posets and Koszul duality. (August 2016)

• Record Type: Journal Article
• Authors: Giraudo, Samuele
• Is Part Of: European journal of combinatorics. Volume 56(2016:Aug.)
• Page Start: 1
• Page End: 32
• Publication Date: 2016-08
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2016.02.008
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: We introduce a functor As from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction As provides a generalization of the associative operad because all of its generating operations are associative. This construction has a very singular property: the operads obtained from As are almost never basic. Besides, the properties of the obtained operads, such as Koszulity, basicity, associative elements, realization, and dimensions, depend on combinatorial properties of the starting posets. Among others, we show that the property of being a forest for the Hasse diagram of the starting poset implies that the obtained operad is Koszul. Moreover, we show that the construction As restricted to a certain family of posets with Hasse diagrams satisfying some combinatorial properties is closed under Koszul duality.