A graded algebra is an associative algebra which is with a labelling on its elements by elements of some monoid or group, and such that the multiplication in the algebra is reflected in the multiplication in the labelling group.
Let be a group. (Often will be abelian, and, in fact, one usually takes by default the additive group of integers, in which case the actual group being used is omitted from the terminology and notation.)
Analogously there is the notion of graded -associative algebra over any commutative ring .
An -graded algebra is called connected if in degree-0 it is just the ground ring
A differential graded algebra is a graded algebra equipped with a derivation of degree +1 (or -1, dependig on conventions) and such that . This is the same as a monoid in the category of chain complexes.
There is a natural isomorphism between
-gradings on ;
-actions on .
Let be any (discrete) group and , its group algebra. This has a direct sum decomposition as a -module,
where each is a one dimensional free -module, for which it is convenient, here, to give a basis . The graded algebra structure is obtained by extending the multiplication rule,
given on basis elements, by -linearity.