Exact sequences, Hochschild cohomology, and the Lie module structure over the $M$-relative center

In this article, we present actions by central elements on Hochschild cohomology groups with arbitrary bimodule coefficients, as well as an interpretation of these actions in terms of exact sequences. Since our construction utilises the monoidal structure that the category of bimodules possesses, we will further recognise that these actions are compatible with monoidal functors and thus, as a consequence, are invariant under Morita equivalences. By specialising the bimodule coefficients to the underlying algebra itself, our efforts in particular yield a description of the degree-$(n,0)$-part of the Lie bracket in Hochschild cohomology, and thereby close a gap in earlier work by S. Schwede.