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In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_ntomathcal{P})[-n]$ for any operad $mathcal{P}$ and a map of operads $fcolon e_ntomathcal{P}$. This result generalizes the result of [T2], where the case $mathcal{P}=mathrm{End}_{Op}(X)$ was considered. Another more computational proof of the same statement was recently sketched in [CW]. Our method combines the one of [T2] with the categorical algebra on the category of symmetric sequences, introduced in [R] and further developed in [KM] and [Fr1]. We define suitable deformation functors on $n$-coalgebras, which are considered as the non-commutative base of deformation, prove their representability, and translate properties of the functors to the corresponding properties of the representing objects. A new point, which makes the method more powerful, is to consider the argument of our deformation theory as an object of the category of symmetric sequences of dg vector spaces, not as just a single dg vector space.
We study the deformation complex of the dg wheeled properad of $mathbb{Z}$-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmu
We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module,
We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra wi
The Structure Theorem for Hopf modules states that if a bialgebra $H$ is a Hopf algebra (i.e. it is endowed with a so-called antipode) then every Hopf module $M$ is of the form ${M}^{mathrm{co}{H}}otimes H$, where ${M}^{mathrm{co}{H}}$ denotes the sp
The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that produ