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Let $D_{poly}$ be the differential graded Lie algebra (dgLA) of differentiable Hochschild cochains on a manifold $\mathscr M$, endowed with the usual Gerstenhaber bracket $[-,-]_G$ and Hochschild diff …

Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$. The assoc …

In his celebrated 97' preprint q-alg/9709040, M. Kontsevich constructs a $L_\infty$-quasi-isomorphism $\mathcal U:\mathcal D_{\rm poly}\to\mathcal T_{\rm poly}$ between the differential graded algebra …

1) In [HIGHER OPERATIONS ON HOCHSCHILD COMPLEX], Gerstenhaber and Voronov showed that the Hochschild complex $C_1(\mathcal A)$ of any associative algebra (or e_1 algebra) $\mathcal A$ is naturally end …

Let $\mathscr P$ be an operad in the category of vector spaces. An algebra (of the type encoded by $\mathscr P$) on the vector space $V$ is a morphism of operads $\mu:\mathscr P\to End_V$ with $End_V$ …

Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following relat …

In his 1997 preprint q-alg/9709040, M. Kontsevich proved constructively the existence of a $L_\infty$-quasi-isomorphism between the differential graded algebra structure on the deformation complex of …

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 wit …

Let $(\mathcal A,\mu)$ be an associative algebra. According to usual deformation theory, deformations of $(\mathcal A,\mu)$ as an associative algebra are controlled by a differential graded algebra (D …

In his 1997 preprint, M. Kontsevich proved the formality of the differential graded algebra controlling deformations of the associative and commutative algebra of functions on a manifold, seen as an a …

Let $\mathfrak g:=(V,d,[\cdot,\cdot])$ be a differential graded Lie algebra (DGLA) where $d$ is the zero differential. Intrinsic formality: The DGLA $\mathfrak g$ will be said intrinsically formal i …

In [Another proof of M. Kontsevich formality theorem], Tamarkin provides a proof of the formality of the differential graded Lie algebra controlling the deformation of a polynomial associative algebra …

As shown by Gerstenhaber and Voronov [Higher operations on the Hochschild complex], the Hochschild complex of an associative algebra is endowed with a natural structure of brace algebra. The first bra …