If $A$ is an associative algebra then central elements of A will act naturally on the derived category $D(A)$. This action allows one (for instance) to build "Koszul complex'' objects in $D(A)$, an extremely important construction in commutative algebra and representation theory.

In a more homotopical situation, when $A$ is a dg algebra, central elements of the homology $H(A)$ may or may not lift to actions on $D(A)$. How can one decide whether this is possible?

In another direction, Buchweitz, Green, Snashall and Soldberg proved that if $A$ is a Koszul algebra (over $k$) then the image of the natural map ${\rm HH}^*(A,A) \to {\rm Ext}^*_A(k,k)$ is exactly the centre of ${\rm Ext}^*_A(k,k)$. For non-Koszul algebras the image will typically be smaller, how can one characterise the image in general? This question comes up in trying to decide whether the so-called "fg-condition'' holds.

These two questions turn out to be essentially the same: the answer to both is the A-infinity centre. I'll talk about this, how it all relates to Koszul duality, and some applications.

This is joint work with Vincent Gélinas.