By J.-L. Loday, A. Frabetti, F. Chapoton, F. Goichot
The most item of analysis of those 4 papers is the proposal of associative dialgebras that are algebras built with associative operations fulfilling a few extra family of the associative variety. This suggestion is studied from a) the homological perspective: building of the (co)homology idea with trivial coefficients and common coefficients, b) the operadic viewpoint: decision of the twin operad, that's the dendriform dialgebras that are strongly similar with the planar binary timber, c) the algebraic viewpoint: Hopf constitution and Milnor-Moore kind theorem.
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Extra resources for Dialgebras and Related Operads
Is associated Its bar-unit, of categories diagram following The entries is denoted Leibniz when D not Proposition. 4. dialgebra For any a new It a to the dialgebras and junctors (3) could is com- mutative Leib Dias T t Lie. As Remark. 5. slightly, In though definition the proposition of a dialgebra, axiom valid. 2 could It be relaxed replaced be by the weaker axiom: ((y Observe the X) I- that -i Z) this in monomials. + (z (X F- formula Hence the -i Y)) = (y variables the I- (X Z)) do not operad associated -i + ((Z stay would X) I- the in be not -1 Y).
One gets ((W ox ((W ox Y) - we = - [[X,Y],Z] [X,Y] +WYO[X,Z] +Wy' Ox' +W' o and Hence ((W i) g =W0[[X,Z],Y] - +WY&[X,Z] ((W x & x W ([X, Y], Z] W Ix, ly, Z11 (W ox) The right and left module structures - - [X, Z], Y]) Wly, Z] +wig ox. 0 9 is immediately are x - 0 x ly,Z]. - U(gLie)-bimodule. The linear map U(gLie) 0 9 -+ U(gLie) W X -+ because w. d). The (nonunital) ideal Of cf. 6. 8 Ud(g) U(gLie)- So, It U(gLie) - (for C T(g) be written 0 9 image and Co its linear as a on 0 Y has one Ud(g)As depends only define can a Co 0 x on U(gLie))- in Indeed, of elements combination w U(gLie)i = of the of follows it E w 0 1.
7. homol- F- y, x D is -d y, x dialgebra. > 1. n sometimes, by DID'. :ftx, x. - J, the m-dimensional =j _irj, P2 ftx, = J, P3 = 1. 7. The erad Dias. 1. (D), d), X, y E D/f = a the By definition dialgebra. 6. copies of DO' (indexed by the by the two products -i, [-, and of cn sum differential complex CY,,(Dias(V)) is proved Its acyclicity We show that it is sufficient to is the Koszul in several treat the complex steps case V as = associated follows. K. (u), complexes split P,,,, is each element for one complex C,,(u) The chain 3.
Dialgebras and Related Operads by J.-L. Loday, A. Frabetti, F. Chapoton, F. Goichot