L-Algebras, triplicial-algebras, within an equivalence of categories motivated by graphs

摘要

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, Δ?_M verifying. In this article, 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) proved in Section 3, the category of L-algebras turns out to be 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 (L-commutative algebras), are briefly evoked and postponed to another article.