What is a Loop?

Date: 10/06/97 at 11:11:04
From: Stephen Gardner
Subject: Loops

This may be a hideously simple question (which will probably make it 
easy to answer), but what is a loop? Further, exactly what is _an_ 
algebra? Thirdly what is a Lie group?

Date: 10/06/97 at 15:37:56
From: Doctor Rob
Subject: Re: Loops

A loop is a set with one operation which is:
  1. closed under the operation,
  2. has a two-sided identity,
  3. has inverses.

A loop with the Associative property would be a group. There is a 
really neat example of a loop of order five with every non-identity 
element of order two.  Here is a multiplication table:

x | 1 a b c d
-------------
1 | 1 a b c d
a | a 1 d b c
b | b c 1 d a
c | c d a 1 b
d | d b c a 1

Loops can be commutative or not.

An algebra can be thought of as a ring with a vector space structure 
over a field. It can also be thought of as a vector space over a field 
on which you define multiplication of vectors. An example is the ring 
of n-by-n matrices over a field. You can add them or multiply them by 
a scalar, and you get the vector space parts of the definition. You 
can also multiply them by each other, and that gives you the ring 
parts of the definition.

A Lie group is a set G such that:
  1. G is a group;
  2. G is a paracompact, real analytic manifold; and
  3. The mapping from G x G --> G defined by (x,y) --> xy^(-1) is real
     analytic.

I don't like this definition much, since I am not a topologist or an
analyst. The simplest example I have found is as follows. Let V be a
finite-dimensional real vector space, and L(V) the associative algebra 
of linear endomorphisms of V. The general linear group GL(V)={x in 
L(V):det(x)<>0} is a Lie group. If you pick a basis of n elements for 
V, then L(V) are the linear transformations from V to V, and GL(V) is 
the group of nonsingular n-by-n real matrices, GL(n,R). The topology 
is the induced topology gotten from the ordinary topology on the real 
numbers.

I'm not sure the last part is very satisfying, but it is a complicated
concept. It was invented to study the relationship between 
differential equations and the groups of operators which preserve 
their solutions.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/