# Group algebras.

Given a group $G$. Let $\mathbb{C}$ be the complex field. Then $\mathbb{C}G$ is the set of linear combinations of elements of $G$. Addition and multiplication are defined as usual. We can also think $\mathbb{C}G$ as the set of all functions from $G$ to $\mathbb{C}$. Addition is the addition of functions and multiplication of two functions is their convolution.

Now if we are give an algebraic group $G$ and $N$ is a subgroup of $G$. I saw the notation in many places. For example, cluster algebras and representation theory. Why $\mathbb{C}[N]$ is called coordinate ring of $N$? What are the elements of $\mathbb{C}[N]$? What are the additions and multiplications in $\mathbb{C}[N]$. Thank you very much.

• nitpick :In my understanding, $\mathbb CG$ is the set of all functions from G to $\mathbb C$ with finite support – gary Aug 6 '11 at 3:33

In that paper $\mathbb C[N]$ denotes the ring of regular functions (also known as the coordinate ring) on the subgroup $N\subset G$, the maximal unipotent subgroup. $N$ is an affine variety, so this makes sense.
You should keep in mind that «we can also think $\mathbb CG$ as the set of all functions from G to $\mathbb C$» only when the group $G$ is finite (and I would add that while you may think that you probably shouldn't) In Leclerc's context this is quite not the case.