Varying the group algebra

A simple mathematical motivation
A group is a nice classical structure -- it has a set of elements, and the product of any two such elements is again an element. The other nice thing about groups is that every group element possesses an inverse.

Now, we would like to vary the group to a quantum setting where the product of two elements need not be an element but a mix of elements. Unfortunately, it does not make sense to mix elements in a purely set-theoretic setting. We need to move to a setting where we can take formal linear combinations. Hence, we work with the group algebra over a field (or sometimes, over a ring).

The group algebra is an algebra with a very distinguished basis -- a basis that is closed under multiplication and inversion. When generalizing the notion, we try to preserve whatever consequences this basis has on the properties of the whole algebra, but remove the assumption of such a distinguished basis.

Between the group algebra and its dual
If $$K$$ is a field and $$G$$ a finite group, there are two algebras of natural interest to us. One is the group algebra, which is noncommutative if $$G$$ is non-Abelian. In other words, the group algebra reflects the noncommutativity of the underlying group.

The other algebra is the algebra of functions on the group, which simply treats the group as a set and defines the multiplication of two functions on the set pointwise. This is a commutative algebra, because multiplication in the base field is commutative.

Incidentally, these algebras are also duals of each other.

One of the goals of varying the group algebra is to find some smooth procedure for moving between the group algebra and its dual, and for studying all the intermediate algebras.

Noncommutative geometry
The idea behind noncommutative geometry is to consider variants of the algebra of functions, which are not commutative, but which we still can view as some sort of algebra of functions. The space on which we imagine those functions to be is the noncommutative space or quantum space.

Group algebra as a Hopf algebra
The group algebra $$H = k(G)$$ can also be given the structure of a coalgebra, by setting the comultiplication as the unique linear map that sends each basis element $$g$$ to $$g \otimes g$$. The counit is defined as the uniqu linear map sending each basis element $$g$$ to $$1$$.

This coalgebra strucutre is compatible with the existing algebra structure, turning the group algebra into a bialgebra.

We can further define an antipode map on the group algebra, by taking the map $$g \mapsto g^{-1}$$ and extending linearly. Under this antipode map, the group algebra satisfies the hypotheses for a Hopf algebra.

Since the comultiplication of the group algebra is cocommutative (in other words, it always gives a symmetric tensor), the group algebra is a cocommutative Hopf algebra. In particular, it is quasitriangular.

Algebra of functions as a Hopf algebra
The algebra of all functions on a group $$G$$, under pointwise multiplication, can also be viewed as a Hopf algebra. Let $$\delta_g$$ denote the function that sends $$g$$ to $$1$$ and sends all the other basis elements to 0. Then