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 $P_1\in H^{\otimes k_1 +1}$ and $P_2\in H^{\otimes k_2+1}$
$$ \label{H-Pre-Lie} P_1\bullet P_2 := \sum_{i=0}^{k_1} (-1)^{ik_2}(\mbox{id}^{\otimes i}\otimes\Delta^{(k_2)}\otimes \mbox{id}^{\otimes k_1-i})(P_1)\cdot (1^{\otimes i}\otimes P_2\otimes 1^{\otimes k_1-i}) $$ and $$ \label{H-Lie} [P_1,P_2]_H := P_1\bullet P_2-(-1)^{k_1k_2}P_2\bullet P_1. $$

This yields the graded Lie algebra $T^\bullet H[1]=\bigoplus_{\bullet=-1}^\infty H^{\otimes \bullet +1}$ with the bracket $[\cdot,\cdot]_H$.

Setting then $m=1\otimes 1\in T^1H[1]$ we find also the differential $\partial_H:=[m,\cdot]_H$. Thus we obtain the DGLA $(H_{poly},\partial_H,[\cdot,\cdot]_H)$.

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 $H_{poly}$ (plus a milder assumption) the element $J=1\otimes 1 +F$ is a Drinfeld twist. Then we can either twist the DGLA $H_{poly}$ by the Maurer-Cartan element to obtain $H^F_{poly}$ or we can twist the Hopf algebra $H$ by the Drinfeld twist (i.e $\Delta_J(X)=J\Delta(X)J^{-1}$) to obtain a DGLA of the twisted algebra $(H_J)_{poly}$. 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 $H_{poly}^0=H$, 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.

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    $\begingroup$ If $A$ is a bialgebra, its cobar construction $\Omega A$ is a brace algebra, see e.g. arxiv.org/abs/math/0406502 and arxiv.org/abs/1309.2820. Your formula for the pre-Lie bracket indeed looks like the formula for the first brace. $\endgroup$ Jul 4, 2017 at 9:04
  • $\begingroup$ Yes! These papers are considering exactly the brace algebra I meant. The only things missing are the canonical Maurer-Cartan element m, or deformation theory etc. It is a good start however. thanks! $\endgroup$ Jul 4, 2017 at 12:56


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