Title 



Differential graded categories and Deligne conjecture
 
Author 


  
Abstract 



We prove a version of the Deligne conjecture for nfold monoidal abelian categories A over a field kk of characteristic 0, assuming some compatibility and nondegeneracy conditions for A . The output of our construction is a weak Leinster (n,1)(n,1)algebra over kk, a relaxed version of the concept of Leinster n algebra in Alg(k)Alg(k). The difference between the Leinster original definition and our relaxed one is apparent when n>1n>1, for n=1n=1 both concepts coincide. We believe that there exists a functor from weak Leinster (n,1)(n,1)algebras over kk to C(En+1,k)C(En+1,k)algebras, welldefined when k=Qk=Q, and preserving weak equivalences. For the case n=1n=1 such a functor is constructed in [31] by elementary simplicial methods, providing (together with this paper) a complete solution for 1monoidal 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)(n,1)algebra over kk, out of an n fold monoidal kklinear abelian category (provided the compatibility and nondegeneracy condition are fulfilled). The second part (still open for n>1n>1) is a passage from weak Leinster (n,1)(n,1)algebras to C(En+1,k)C(En+1,k)algebras. As an application, we prove in Theorem 8.1 that the GerstenhaberSchack complex of a Hopf algebra over a field kk of characteristic 0 admits a structure of a weak Leinster (2,1)(2,1)algebra over kk extending the Yoneda structure. It relies on our earlier construction [30] of a 2fold monoidal structure on the abelian category of tetramodules over a bialgebra.   
Language 



English
 
Source (journal) 



Advances in mathematics.  New York, N.Y.  
Publication 



New York, N.Y. : 2016
 
ISSN 



00018708
 
Volume/pages 



289(2016), p. 797843
 
ISI 



000369123100021
 
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