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

The most item of research of those 4 papers is the suggestion of associative dialgebras that are algebras outfitted with associative operations gratifying a few extra kinfolk of the associative variety. This concept is studied from a) the homological perspective: building of the (co)homology conception with trivial coefficients and normal coefficients, b) the operadic viewpoint: decision of the twin operad, that's the dendriform dialgebras that are strongly comparable with the planar binary timber, c) the algebraic perspective: Hopf constitution and Milnor-Moore variety theorem.

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Extra info for Dialgebras and Related Operads

Example text

Vn) For instance V2f)IV3 V2 V3 + V3V2f)l- One observes [Lo] is -/j - V2) (D described are in by vn ... symbol (VI As(V) T(V) Lie(V) describe that the Proposition. algebra on this map is very of loop suspension Let V be Then V. one Ud(Leib(V)) --- a wedge product K-module has an Dias(V) to the map similar a used by topological a of Leib(V) and = T(V) Lodder in spaces. be the free isomorphism T(V) = T(V). 0 V0 from Leibniz functor to the forgetful The functor Leib is left adjoint Proof. to the functor Ud is left the functor adjoint Similarly algebras to modules.

Obtained follows. as Let = Start ... with of the ... of numbers largest [a, integer ... in ai aj [ai+,-... ai+k+l ai+kl). 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.

0] and >- described Proof. [0] Y this following follow equalities associativity and the (i) (Y of -< YI) Y1 V = (Y us Y YI V Y2 be * >- Y, * a Proposition. Let us The show that the universal satisfies y" the (total) y2" the be operations the two by [1] =\\Y/. generated is V from with which on yj" = (Y2 = V planar definitions degree of the trees. binary trees. The of the operations := [11 Y11) f / (YI + Y -< Yfl)- >- ff YIif) 2 *Y ff) if) V Y2 y f) y/I >_ (Y + yif >_y --< and by t1] under the operations of the operations we have >-.