The anticyclic operad of moulds

A new anticyclic operad Mould is introduced, on spaces of functions in several variables. It is proved that the Dendriform operad is an anticyclic suboperad of this operad. Many operations on the free Mould algebra on one generator are introduced and studied. Under some restrictions, a forgetful map from moulds to formal vector fields is then defined. A connection to the theory of tilting modules for quivers of type A is also described. 0 Introduction The aim of this article is to build and use a new connection between the theory of operads and the theory of moulds. Operads were introduced in algebraic topology in the 1960’s. After being somewhat neglected for some decades, this notion has found a new impetus recently, in connection with mathematical physics, moduli spaces of curves and algebraic combinatorics. Moulds have a rather different origin. They have been introduced in analysis by J. Ecalle, as a convenient tool to handle complicated singular functions, in relation with his theory of resurgence. Later, he developed around moulds a large apparatus which allowed him to make substantial progress in the theory of polyzetas [Eca02, Eca03, Eca04]. In this article, only the simplest case of moulds will be considered, not the more general case of bimoulds. The first result of this article is the existence of a very simple structure of operad on moulds, denoted by Mould. In fact, one can define on moulds the finer structure of an anticyclic operad, involving in addition to composition maps some actions of cyclic groups. This structure is then shown to contain as an anticyclic suboperad the socalled Dendriform operad introduced by J.-L. Loday [Lod01] and denoted by Dend, which has been much studied recently [LR98, LR02]. This provides a radically new point of view on the operad Dend and the key to some new results. The main result is the explicit description of the smallest subset of Dend containing its usual generators and closed under the anticyclic operad structure, by the mean of new combinatorial objects called non-crossing plants. This article also contains the description of many different operations on moulds, coming either from the operad or from the mould viewpoint, and some of their properties. This is used to prove the existence of a morphism from the Lie algebra of moulds (for one of the Lie brackets and under some restrictions) to the Lie algebra of formal vector fields in one indeterminate. Interesting and