Coassociative Magmatic Bialgebras and the Fine Numbers

We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n − 2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number Fn−1. In short, the triple of operads (As, Mag,MagF ine) is good. Introduction A magmatic algebra is a vector space equipped with a unital binary operation, denoted x · y with no further assumption. Let us suppose that it is also equipped with a counital binary cooperation, denoted ∆(x). There are different compatibility relations that one can suppose between the operation and the cooperation. Here are three of them: Hopf: ∆(x · y) = ∆(x) ·∆(y) magmatic: ∆(x · y) = x · y ⊗ 1 + x⊗ y + 1⊗ x · y unital infinitesimal: ∆(x · y) = ∆(x) · (1⊗ y) + (x⊗ 1) ·∆(y)− x⊗ y On the free magmatic algebra the cooperation ∆ is completely determined by this choice of compatibility relation and the assumption that the generators are primitive. In the Hopf case ∆ is coassociative and cocommutative, giving rise to the notion of Com-Mag-bialgebra. This case has been addressed in [7]. In the magmatic case ∆ is comagmatic, giving rise to the notion of Mag-Mag-bialgebra. This case has been addressed by E. Burgunder in [2]. In the u.i. case, ∆ is coassociative, giving rise to the notion of As-Mag-bialgebra. This case is the subject of this paper. First, we construct a functor F : Mag-alg → MagFine-alg, whereMagFine is the operad of algebras having (n − 2) operations of arity n for n ≥ 2 (no binary operation, one ternary operation, etc.). Then we show that the primitive part of any As-Mag-bialgebra is closed under theMagFine operations. The functor F has a left adjoint denoted U : MagFine-alg → Mag-alg. Date: February 2, 2008. 2000 Mathematics Subject Classification. 16A24, 16W30, 17A30, 18D50, 81R60.