The A-infinity Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category

Benjamin Briggs; Vincent Gelinas

paper

The A-infinity Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category

arXiv:1702.00721

Abstract

For A a dg (or A-infinity) algebra and M a module over A, we study the image of the characteristic morphism $χ_{M} : H H^{*} (A, A) \to E x t_{A} (M, M)$ and its interaction with the higher structure on the Yoneda algebra $E x t_{A} (M, M)$ . To this end, we introduce and study a notion of A-infinity centre for minimal A-infinity algebras, agreeing with the usual centre in the case that there is no higher structure. We show that the image of $χ_{M}$ lands in the A-infinity centre of $E x t_{A} (M, M)$ . When A is augmented over k, we show (under mild connectedness assumptions) that the morphism $χ_{k} : H H^{*} (A, A) \to E x t_{A} (k, k)$ into the Koszul dual algebra lands exactly onto the A-infinity centre, generalising the situation from the Koszul case established by Buchweitz, Green, Snashall and Solberg. We give techniques for computing A-infinity centres, hence for computing the image of the characteristic morphism, and provide worked-out examples. We further study applications to topology. In particular we relate the A-infinity centre of the Pontryagin algebra to a wrong way map coming from the homology of the free loop space, first studied by Chas and Sullivan.

43 pages. Comments welcome