Lie Groups

A manifold, to back up a step, is merely a mathematically continuous geometrical space, such as a curved line on a graph (in two dimensional terms), the surface of a sphere (three dimensions), or even the shape of space-time itself (four dimensions).

While manifolds can exist in a great many forms with very few limitations, for a manifold to exist as part of a lie group, it must pass the test of being differentiable.

Differentiability and Lie Groups

In calculus, to say that a given object is differentiable simply means that it is possible to create an equation (or system of equations) which is fully self-consistent in describing what is "happening" at any point within a continuous manifold.

If one had a simple graph of a curved line, then, simple calculus would lead one to establish a differential equation which would demonstrate the exact "rate of change" at any given point on the line - whether it is rising or falling, and how quickly.

If a given manifold of any number of dimensions (modern geometers and physicists are familiar with lie groups of many dimensions) is differentiable, then it is part of a lie group, which is simply a method of categorizing these various manifolds.

Using of Lie Groups

One of the first mathematicians to begin to develop methods by which to work with geometrical manifolds was, naturally, the one for which these groups were named - Norwegian mathematician Sophus Lie.

It is not merely the grouping of manifolds together or the categorization of them in which lie groups find their greatest mathematical importance. Rather, it is the methods (which are still being formulated and worked out to this day, as this is a very complicated and not yet perfect form of mathematics) by which transformations and operations may be performed within these groups.

Methods by which differential calculus may be used to work within different forms of geometry (that is, different topological forms of non-Euclidean geometry) have become known as Lie Algebra.

The systems and methods developed within this branch of mathematics have become absolutely crucial, not just from the standpoint of pure mathematics, but perhaps even more so from the perspective of theoretical physics, where Lie Groups play an important role in several areas, including General Relativity and Quantum Mechanics, where finding new and more efficient ways of working with infinitesimal transformations is one of the keys to success.

While much of the information which could be said about Lie Groups would surely fly far over the heads of all but the most devoted students of mathematicians, what should be recognized is their importance in the continually growing study of geometry and the nature of space and time itself.

References:

Lie Group. Wolfram's Mathworld.

A diagram showing a warping space-time manifold

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