In a seminal paper "On the Deformation of Rings and Algebras", M. Gerstenhaber showed that the deformation complex of any associative algebra (known as the Hochschild complex) is naturally endowed with a structure of (later called) Gerstenhaber algebra (consisting of a degree 0 commutative associative product together with a compatible degree -1 Lie bracket).

The notion of Gerstenhaber algebra can be naturally extended to one of a $\hat p$-Gerstenhaber algebra where the Lie bracket is of degree $1-\hat p$.

$1$-Gerstenhaber algebras are associative algebras.

$2$-Gerstenhaber algebras are usual Gerstenhaber algebras.

In math/0010072, Tamarkin generalised the result of Gerstenhaber by showing that the cohomology of the deformation complex of a $\hat p$-Gerstenhaber algebra is naturally a $\hat p+1$-Gerstenhaber algebra (rather he showed that there is a homotopy $\hat p+1$-Gerstenhaber algebra structure on the deformation complex itself).

In particular, the cohomology of the deformation complex of a Gerstenhaber algebra is naturally a $3$-Gerstenhaber algebra.

Question: Are there other known structures whose deformation complex admits a $3$-Gerstenhaber algebra structure on cohomology?


1 Answer 1


Here are two deformation complexes which have a natural 3-Gerstenhaber (aka $\mathbb{P}_3$) algebra structure:

  1. Deformations of a Lie bialgebra $\mathfrak{g}$ are controlled by the complex $\mathrm{Sym}(\mathfrak{g}[-1]\oplus\mathfrak{g}^*[-1])$. It has a bracket uniquely determined on the generators to be the obvious pairing (the so-called 'big bracket'). Under Koszul duality between Lie bialgebras and Gerstenhaber algebras this deformation complex goes to the deformation complex of the corresponding Gerstenhaber algebra.

  2. Deformations of a bialgebra $A$ are controlled by the Gerstenhaber--Schack complex $\bigoplus_{n,m}\mathrm{Hom}(A^{\otimes n}, A^{\otimes m})$. It has an $\mathbb{E}_3$-algebra structure (https://arxiv.org/abs/1606.01504), i.e. it's an algebra over the operad of little 3-disks. So, its cohomology is a $\mathbb{P}_3$-algebra.

One can obtain quantizations of Lie bialgebras from an $L_\infty$ quasi-isomorphism between these two complexes in the case $A=\mathrm{Sym}(\mathfrak{g})$ (see e.g. the Ginot--Yalin paper).


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