Se p 20 07 An equivalence of categories motivated by weighted directed graphs

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, ∆M , ∆̃M verifying, (∆̃M ⊗ id)∆M = (id⊗∆M)∆̃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 triplicialalgebras: 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. 1 An algebraic setting over weighted directed graphs In the sequel, K will be a characteristic zero field and its unit will always be denoted by 1K . The symmetric group over n elements is denoted by Sn and if P denotes a regular operad, then we write P(n) = Pn ⊗KSn, the K-vector space of the n-ary operations of P, see for instance [19] for notation and basic definitions in operad theory. The symbol ◦ 1 2000 Mathematics Subject Classification: 05E99, 05C20, 05C05, 16W10, 16W30, 17A30, 17A50, 18D50, 60J99.