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

TL;DR

This paper investigates the A-infinity centre of the Yoneda algebra and how Hochschild cohomology acts on the derived category, introducing new notions and providing computational techniques with applications to topology.

Contribution

It introduces the concept of A-infinity centre for minimal A-infinity algebras and shows its relation to the characteristic morphism and Hochschild cohomology.

Findings

The image of the characteristic morphism lands in the A-infinity centre.

Under mild conditions, the morphism lands exactly onto the A-infinity centre.

Provides techniques and examples for computing A-infinity centres.

Abstract

For A a dg (or A-infinity) algebra and M a module over A, we study the image of the characteristic morphism $\chi_M: HH^*(A, A) \to Ext_A(M, M)$ and its interaction with the higher structure on the Yoneda algebra $Ext_A(M, M)$. 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 $\chi_M$ lands in the A-infinity centre of $Ext_A(M, M)$. When A is augmented over k, we show (under mild connectedness assumptions) that the morphism $\chi_k: HH^*(A, A) \to Ext_A(k,k)$ 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.