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/ |
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