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تسجيل مستخدم جديدGraded Leinster monoids and generalized Deligne conjecture for 1-monoidal abelian categories
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تأليف
Boris Shoikhet
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In our recent paper [Sh1] a version of the generalized Deligne conjecture for abelian $n$-fold monoidal categories is proven. For $n=1$ this result says that, given an abelian monoidal $k$-linear category $mathscr{A}$ with unit $e$, $k$ a field of characteristic 0, the dg vector space $mathrm{RHom}_{mathscr{A}}(e,e)$ is the first component of a Leinster 1-monoid in $mathscr{A}lg(k)$ (provided a rather mild condition on the monoidal and the abelian structures in $mathscr{A}$, called homotopy compatibility, is fulfilled). In the present paper, we introduce a new concept of a ${it graded}$ Leinster monoid. We show that the Leinster monoid in $mathscr{A}lg(k)$, constructed by a monoidal $k$-linear abelian category in [Sh1], is graded. We construct a functor, assigning an algebra over the chain operad $C(E_2,k)$, to a graded Leinster 1-monoid in $mathscr{A}lg(k)$, which respects the weak equivalences. Consequently, this paper together with loc.cit. provides a complete proof of the generalized Deligne conjecture for 1-monoidal abelian categories, in the form most accessible for applications to deformation theory (such as Tamarkins proof of the Kontsevich formality).
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We prove a version of the Deligne conjecture for $n$-fold monoidal abelian categories $A$ over a field $k$ of characteristic 0, assuming some compatibility and non-degeneracy conditions for $A$. The output of our construction is a weak Leinster $(n,1
)$-algebra over $k$, a relaxed version of the concept of Leinster $n$-algebra in $Alg(k)$. The difference between the Leinster original definition and our relaxed one is apparent when $n>1$, for $n=1$ both concepts coincide. We believe that there exists a functor from weak Leinster $(n,1)$-algebras over $k$ to $C(E_{n+1},k)$-algebras, well-defined when $k=mathbb{Q}$, and preserving weak equivalences. For the case $n=1$ such a functor is constructed in [Sh4] by elementary simplicial methods, providing (together with this paper) a complete solution for 1-monoidal abelian categories. Our approach to Deligne conjecture is divided into two parts. The first part, completed in the present paper, provides a construction of a weak Leinster $(n,1)$-algebra over $k$, out of an $n$-fold monoidal $k$-linear abelian category (provided the compatibility and non-degeneracy condition are fulfilled). The second part (still open for $n>1$) is a passage from weak Leinster $(n,1)$-algebras to $C(E_{n+1},k)$-algebras. As an application, we prove that the Gerstenhaber-Schack complex of a Hopf algebra over a field $k$ of characteristic 0 admits a structure of a weak Leinster (2,1)-algebra over $k$ extending the Yoneda structure. It relies on our earlier construction [Sh1] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra.
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