The idea behind noncommutative geometry is that one starts with
a

noncommutative algebra *A*, lets say, and

**imagines** that it is the

coordinate ring
of an

*noncommutative* space. The space doesn't really
exist, we just

**act** as though it does.
It turns out that there are several classes
of algebras for which this idea is natural and leads to useful
results in

theoretical physics.

Let's have an example. Consider the plane *k*^{2},
for an algebraically closed field *k*(such as the complex numbers).
This has coordinate ring the polynomial ring in two variables
*k[x,y]*. So that's the usual commutative setup. We
are going to deform this by a nonzero parameter *q* in *k*.
Thus define *k*_{q}[x,y] to be the *k*-algebra
generated by two variables *x,y* and subject to the relation
(see generators and relations for algebras for the precise meaning of this)

*
xy = qyx
*

If we take

*q=1* then we get back the usual

polynomial ring
in two variables but in all other cases we have a noncommutative
ring. We think of this as the coordinate ring of a

**quantum** plane.
So already we see something interesting, there are several quantum planes
around, whereas there is just one commutative one.
Here's something else to think about.

Hilbert's Nullstellensatz
tells us that the

maximal ideals of

*k[x,y]*
are in one-one correspondence with the points of the commutative plane

*k*^{2}. So we might think that the maximal ideals
of

*k*_{q}[x,y] as representing the points
of the quantum plane. If

*q* is not a root of unity then
it's quite easy to see that the maximal ideals have the form

*(y,x-a)* or

*(y-b,x)*, for

*a,b* in

*k*.
In other words the quantum plane has far fewer points than thus usual
plane, it just has the union of the two coordinate axes.

One place where noncommutative geometry has been very successful
is in the theory of **quantum groups**. As you can probably guess
from the above quantum groups are not really groups at all. They
are imaginary noncommutative Lie groups. The concrete objects
that actually exist are certain noncommutative rings which we imagine to
be coordinate rings, just as above. Here's a specific example
it the coordinate ring of a quantum group which is a deformation of
SL_{2}. Again there is a family
of deformations associated to a nonzero parameter *q* in *k*.
One takes the algebra generated by *a,b,c,d* subject to the relations

*
ab = q*^{-1}ba

ac = q^{-1}ca

cd = q^{-1}dc

bd = q^{-1}db

bc = cb

ad - da = (q^{-1}-q)bc

ad - q^{-1}bc = 1

The last relation sets the quantum determinant to 1.
It turns out that quantum groups turn up in a few different
places. In

physics they are related to solutions of
the

Yang-Baxter equation. In

mathematics they are related to

knot theory.