L-algebras, triplicial-algebras, within an equivalence of categories motivated by graphs
Abstract
Description
In a previous work, we gave a coalgebraic framework of directed graphs equipped with weights (or probability vectors) in terms of (Markov) L-coalgebras. They are K-vector spaces equipped with two co-operations, Δ_M, \tildeΔ_M verifying,
(\tildeΔ_M \otimes id)Δ_M =(id \otimes Δ_M)\tildeΔ_M.
In this paper, we study the category of L-algebras (dual of L-coalgebras), prove that the free L-algebra on one generator is constructed over rooted planar symmetric ternary trees with odd numbers of nodes and the L-operad is Koszul. We then introduce triplicial-algebras: vector spaces equipped with three associative operations verifying three entanglement relations. The free triplicial-algebra is computed and turns out to be related to even trees. Via a general structure theorem (à la Cartier-Milnor-Moore), we obtain that the category of L-algebras is equivalent to a much more structured category called connected coassociative triplicial-bialgebras (coproduct linked to operations via infinitesimal relations), that is the triple of operads (As, Trip, L) is good. Bidirected graphs, related to NAP-algebras, are briefly evoked and postponed to another paper.
34 pages, 5 figures. A new presentation of the free triplicial-algebra has been added (Subsection 5.5) + modification of the proof of Thm 5.1. The title has been changed too
34 pages, 5 figures. A new presentation of the free triplicial-algebra has been added (Subsection 5.5) + modification of the proof of Thm 5.1. The title has been changed too