Thursday, October 27, 2005

Groups, rings, modules, vector spaces, Hilbert spaces

Consider a set of elements X= {a,b,c,...}and an operator *.

Group Theory requires four axioms. They are:
i) closure law: for any a and b in X, then a*b is also in X,
ii) associative law: a*(b*c) = (a*b)*c,
iii) Identity law: there is an element e such that a*e = e*a = a,
iv) Inverse law: for any a there is an element a-1 such that a * a-1= a-1*a = e. (For ordinary multiplication, the element a cannot be equal to zero).

If a group obeys the following rule: a*b = b*a then it is said to be commutative or abelian.

A ring has two operators such as * and +.
In addition to the above four axioms for each of the two operators, it has a distributive law:
a*(b+c) = a*b+a*c.

This is the essence of high school algebra.

A module A is like a set of primitive vectors. They are multiplied by a suitable ring R of scalars.

Consider a module A= {a,b,c...} and a ring R= {k,l,m...}
Definition: An R-module A is an additive commutative group together with a function that maps (k,a) into ka, subject to the following axioms:
i) k(a+b) =ka+kb
ii) (k+m)a= ka+ma
iii) (km)a=k(ma)
iv) ea=a, where e is the identity element

More explicitly, such a module is a left module because in forming ka the scalar k is written on the left of the module a. One can re-define a right module using the same procedure to define ak, with the scalar on the right.

Now if the ring is the whole field of real numbers or the complex numbers, then the module is a vector space.

Now a Hilbert space is defined as previously mentioned, with a slightly different language.(See Hilbert Space and Quantum Mechanics). Every inner product (.,.) between two vectors on a real or complex vector space H gives rise to a norm or the length of the vector, as follows:

x = sqrt (x,x)

We say that H is a Hilbert space if it is complete with respect to this norm. Completeness in this context means that any sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero.

4 comments:

Gustavo K-fé said...

What if the ring is not a field, but only a commutative ring ( Z or Z12 for instance)? Can I still find an inner product?

joseph palazzo said...

There is a minimum of ring multiplications necessary to compute the inner product of two n-vectors over a noncommtattive ring.

See http://locus.siam.org/SICOMP/volume-04/art_0204004.html

yaron said...

The definition of RING wasn't accurate. Not the both operation should satisfy the group axioms.

joseph palazzo said...

Technically, you are right. The second operation forms a monoid with identity 1:
a*1=1*a=a
a*(b*c)=(a*b)*c

There is no inverse law on account that one can get division by zero.