Are G_infinity algebras B_infinity? Vice versa? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:15:20Z http://mathoverflow.net/feeds/question/20453 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20453/are-g-infinity-algebras-b-infinity-vice-versa Are G_infinity algebras B_infinity? Vice versa? Ian Shipman 2010-04-06T00:56:25Z 2010-04-06T17:59:42Z <p>What is the relationship between $G_\infty$ (homotopy Gerstenhaber) and $B_\infty$ algebras?</p> <p>In Getzler &amp; 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 $m$. </p> <p>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?</p> <p>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? </p> http://mathoverflow.net/questions/20453/are-g-infinity-algebras-b-infinity-vice-versa/20530#20530 Answer by Ben for Are G_infinity algebras B_infinity? Vice versa? Ben 2010-04-06T17:59:42Z 2010-04-06T17:59:42Z <p>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.</p>