A very cool branch of Mathematics which is somewhat motivated by Quantum Mechanics. There are all these cool theorems connecting Commutative Algebra and Geometry or Topology and Quantum Mechanics provides a little bit of motivation about how&to do simillar things with Non-Commutative Algebra.

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 k2, 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 kq[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 k2. So we might think that the maximal ideals of kq[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 SL2. 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-1ba
ac = q-1ca
cd = q-1dc
bd = q-1db
bc = cb
ad - da = (q-1-q)bc
ad - q-1bc = 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.

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