  We propose a generalisation for the notion of the centre of an algebra in the
setup of algebras graded by an arbitrary abelian group G.
  Our generalisation, which we call the G-centre, is designed to control the
endomorphism category of the grading shift functors. We show that the G-centre
is preserved by gradable derived equivalences given by tilting modules. We also
discuss links with existing notions in superalgebra theory and apply our
results to derived equivalences of superalgebras.
