symmetric monoidal (∞,1)-category of spectra
A Gerstenhaber algebra is a Poisson 2-algebra, a Poisson algebra in graded vector spaces with Poisson bracket of degree -1.
A Gerstenhaber algebra is a chain complex $A$ equipped with
a symmetric product $\cdot : A \otimes A \to A$;
a skew-symmetric bracket $[-,-] : A \otimes A \to A[1]$;
such that associativity of $\cdot$ and the Jacobi identity for $[-,-]$ holds and such that $[a,-]$ is a derivation of $\cdot$.
The homology of the operad for Gerstenhaber algebras in chain complexes is the operad for Gerstenhaber algebras.
Accordingly the homology of an E2-algebra is a Gerstenhaber algebra.
This is due to Cohen (1976).
A Gerstenhaber algebra equipped in addition with a certain morphism $\Delta : A \to A$ is a BV-algebra. This is the homology of an algebra over the framed little 2-disk operad.
Cohen (1976)
Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math., 78 (1963), 267-288 MR28:5102
Ping Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Commun. Math. Phys. 200, No.3, 545-560 (1999).
Last revised on December 21, 2016 at 10:01:21. See the history of this page for a list of all contributions to it.