Dialgebras and Related Operads by J.-L. Loday, A. Frabetti, F. Chapoton, F. Goichot

By J.-L. Loday, A. Frabetti, F. Chapoton, F. Goichot

The most item of analysis of those 4 papers is the suggestion of associative dialgebras that are algebras built with associative operations gratifying a few extra relatives of the associative style. This proposal is studied from a) the homological standpoint: development of the (co)homology thought with trivial coefficients and common coefficients, b) the operadic perspective: choice of the twin operad, that's the dendriform dialgebras that are strongly comparable with the planar binary bushes, c) the algebraic standpoint: Hopf constitution and Milnor-Moore sort theorem.

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2 we get a --+ functor Dend Summarizing we Dend. - ++ As. get the following The following Proposition. 4. Zinb Com diagram of functors Dend As between categories of 47 Proof. Y+Y-X=XY Theorem. 5. For Dend(V) algebras K[Y,,] (2) vector any K[Sn] is K[Sn], -+ the V, induced, space Zinb(V)Dend -+ of dendriform by the map natural degree in map n, E y & w -+ aw , JOIESn I 0'(a)=Y} where 0' : Sn-*Yn is the map described surJective in Appendix A6. observe that the map V -+ Zinb(V) gives a map (the same on First, Proof.

An] In into our a name example of tree we as get [131]. before (aj is the 42 Proposition. 11. 7 induces compo- operation in (Dn>,K[Yn]: Y where the sub-tree Proof. distinct = it suffices operad is quadratic x --< isomorphism gives [21] of composition are: (x --< x) x = x [21] ol [12] = (x >- x) x = [1311 [21] 02 [21] = x (x = [321] [21] 02 [12] = x (x [12] ol [21] = (x [12] ol [12] = (x [12] 02 [21] = x - (x --< [12] 02 [12] = x >- (x >- we case >- verify is nested x) = [312] [213] x) >- x x) >- x = [123] x) = [131] x) = x that vector -< as nested a this check [12] and = assertion >- x in The x.

4. 2n, dual, we use is we recover (up and onn 1 ... I by n). , 1). The map r CZinb(R) CDend(RDend) of the Xn) On (X1, explicit induced by the morphism of of the description Appendix Dias(V)Leib, of the signed action of (n) permutations. the 2' monomials of [xi [X2) permutation) sum to the n), (r, + (1, i recursively 135e. Since r Xn111 , 0 KOSZULDUALITY FORTHE DIALGEBRA OPERAD In this is (r, n' (1,... cf. -+ 0'n is the defined by (xi, the chain complex map 6* : operad morphism Dend -+ ZZnb. This = n) is Proof.

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