DGLA related to the deformation of hopf Algebras

Recently I was considering Hopf algebras and Drinfeld’s twists. I stumbled upon
a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by looking at the universal enveloping algebra of a Lie algebra(oid) and the Gerstenhaber structure induced by viewing them as (left invariant) polydifferential operators.

Explicitely, given a hopf algebra H we set for
P1Hk1+1 and P2Hk2+1
P1P2:=k1i=0(1)ik2(idiΔ(k2)idk1i)(P1)(1iP21k1i)
and
[P1,P2]H:=P1P2(1)k1k2P2P1.

This yields the graded Lie algebra TH[1]==1H+1 with the bracket [,]H.

Setting then m=11T1H[1] we find also the differential H:=[m,]H. Thus we obtain the DGLA (Hpoly,H,[,]H).

Essentially my question is for a reference for a discussion of this DGLA or some other places it shows up. For a little more context let me mention the following. Given a Maurer-Cartan element F of Hpoly (plus a milder assumption) the element J=11+F is a Drinfeld twist. Then we can either twist the DGLA Hpoly by the Maurer-Cartan element to obtain HFpoly or we can twist the Hopf algebra H by the Drinfeld twist (i.e ΔJ(X)=JΔ(X)J1) to obtain a DGLA of the twisted algebra (HJ)poly. It turns out that these two are canonically isomorphic (the isomorphism is canonically determined by J). Also, the DGLA structure above is as defined coming from a pre-Lie structure. It is not hard to see that in fact this pre-Lie structure is coming from a brace algebra structure (again by the formulas as they appear for polydifferential operators). In searching for a reference related to this DGLA I did find a mention of the induced brace algebra structure on H0poly=H, namely in https://arxiv.org/pdf/math/0211074.pdf . But no mention of the larger brace algebra.

I hope the question is not too vague and someone has seen this creature somewhere before.

Answer

Attribution
Source : Link , Question Author : Niek de Kleijn , Answer Author : Community

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