Differential graded categories and Deligne conjecture

作者：Shoikhet Boris^*

来源：Advances in Mathematics, 2016, 289: 797-843.

DOI：10.1016/j.aim.2015.11.030

• Higher-monoidal categories
• Differential graded categories
• dg quotient
• Deformation theory
• Deligne conjecture

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, *algebra over k, a relaxed version of the concept of Leinster n -algebra in illg(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, *algebras over k to C. (En+1, k) -algebras, well-defined when k = Q, and preserving weak equivalences. For the case n = 1 such a functor is constructed in [31] 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, *algebra over out of an n -fold monoidal 1k -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, *algebras to C. (En+1, k) -algebras. As an application, we prove in Theorem 8.1 that the Gerstenhaber Schack complex of a Hopf algebra over a field Ik of characteristic 0 admits a structure of a weak Leinster (2, 1)-algebra over 1k extending the Yoneda structure. It relies on our earlier construction [30] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra.

• 出版日期2016-2-5