SL(n,K) and U(n,K): an Algebraic Geometry Approach

I considered the singular matrix as an algebraic variety before.

I observed another interesting example today.

Let us consider the special linear group over algebraic closed fields $K$$K$.

$$\begin{array}{}\text{(1)}& SL(n,K):=\{X\in GL(n,K)|detX=1\}\end{array}$$

Which is a variety since it is the zero of $detX-1=0$$\det X-1=0$

Remark

$$\begin{array}{}\text{(2)}& detX\in K[X_{1,1},...,X_{n,n}]\end{array}$$

and

$$\begin{array}{}\text{(3)}& SL(n,K)\in {\mathbb{A}}_K^{n^2}\end{array}$$

Math Cat told me that this is a kind of algebraic group, i.e. group object in the category of variety.

$SL(n,K)$$SL(n,K)$ has a sub-variety, $U(n,K)$$U(n,K)$.

Which is the intersection of $X_{1j}^2+X_{2,j}^2+...+X_{n,j}^2-1=0$$X_{1j}^2+X_{2,j}^2+...+X_{n,j}^2-1=0$, $j$$j$ from $1$$1$ to $n$$n$.

One interesting question is what is the property of the coordinate ring of $SL(n,K)$$SL(n,K)$ and $U(n,K)$$U(n,K)$?