A Hopf algebra is a ring with some additional structure. Many important examples arising in various fields in mathematics include group algebras, universal enveloping algebras of Lie algebras, and quantum groups. Quantum groups are not groups, but instead are Hopf algebras that share some essential properties with groups. Groups arise in the study of symmetry, such as that of atoms and molecules, or of geometric spaces or of solutions of equations. Quantum groups have their origins in statistical mechanics and knot theory, and they as well as the more general Hopf algebras also can be regarded as exposing symmetry. One important property of modules (or representations) for a Hopf algebra is that the tensor product of two modules, over a commutative ground ring, is again a module. This property gives its category of modules the structure of a tensor category, which is very beneficial for understanding its modules and their cohomology.

A group-graded ring is a ring that decomposes into a direct sum of components indexed by elements from a group in such a way that multiplication follows the group multiplication. Some important examples of such algebras arise in geometry: If a group acts on a space (such as a manifold or variety), one may form an algebra from the group and an algebra of functions on the space. This algebra is graded by the group and it encodes information about the space. One is interested in cohomology and deformations of such algebras in relation to cohomology and deformations of the original spaces. Information about deformations of the algebras is contained in their Hochschild cohomology. Some important examples of their deformations are rings that were discovered independently in several different contexts, termed graded (Drinfeld) Hecke algebras, symplectic reflection algebras, and rational Cherednik algebras.