What is the relationship between $G_\infty$ (homotopy Gerstenhaber) and $B_\infty$ algebras?

In Getzler & Jones "Operads, homotopy algebra, and iterated integrals for double loop spaces" (a paper I don't well understand) a $B_\infty$ algebra is defined to be a graded vector space $V$ together with a dg-bialgebra structure on $BV = \oplus_{i \geq 0} (V[1])^{\otimes i}$, that is a square-zero, degree one coderivation $\delta$ of the canonical coalgebra structure (stopping here, we have defined an $A_\infty$ algebra) and an associative multiplication $m:BV \otimes BV \to BV$ that is a morphism of coalgebras and such that $\delta$ is a derivation of $m$.

A $G_\infty$ algebra is more complicated. The $G_\infty$ operad is a dg-operad whose underlying graded operad is free and such that its cohomology is the operad controlling Gerstenhaber algebras. I believe that the operad of chains on the little 2-discs operad is a model for the $G_\infty$ operad. Yes?

It is now known (the famous Deligne conjecture) that the Hochschild cochain complex of an associative algebra carries the structure of a $G_\infty$ algebra. It also carries the structure of a $B_\infty$ algebra. Some articles discuss the $G_\infty$ structure while others discuss the $B_\infty$ structure. So I wonder: How are these structures related in this case? In general?


1 Answer 1


There is a nice summary of the relationship between B infinity and G infinity in the first chapter of the book "Operads in Algebra, Topology and Physics" by Markl, Stasheff and Schnider. The short answer is G infinity is the minimal model for the homology of the little disks operad (the G operad). B infinity is an operad of operations on the Hochschild complex. Many of the proofs of Deligne's conjecture involve constructing a map between these two operads.

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    $\begingroup$ More precisely, there is an opeard map (constructed by Tamarkin, and depending on the choice of an associator) $G_\infty\to B_\infty$. In other words, any $B_\infty$-algebra is a $G_\infty$-algebra. To see this, one has to remember that a $G_\infty$-structure can be expressed in terms of a DG Lie bialgebra structure on a cofree Lie coalgebra. On the other hand a $B_\infty$-algebra structure can be expressed in terms of DG bialgebra structure on a cofree coassociative algebra. the relation between the two is given by Etingof-Kazhdan dequantization Theorem. $\endgroup$
    – DamienC
    Apr 25, 2011 at 21:14

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