## What is a group?Mathematicians invented the concept of agroup to capture the
essence of symmetry. The collection of symmetries of any object is a
group, and every group is the symmetries of some object.
E is a rather complicated group: it is the symmetries of a particular
57 dimensional object, and _{8}E itself is 248 dimensional!
_{8}E is even more special: it is a _{8}Lie group,
which means that it also has
a nice geometric structure.
The theory of groups has found widespread application. It was used to determine the possible structure of crystals, and it has deep implications for the theory of molecular vibration. The conservation laws of physics, such as conservation of energy and conservation of electric charge, all arise from the symmetries in the equations of physics. And one of the simplest groups, known as "the multiplicative group modulo N" is used every time you send secure information over the Internet. For an introduction to groups requiring little mathematics background, see Groups and Symmetry by David Farmer. ## What is a Lie group?Lie groups lie at the intersection of two fundamental fields of mathematics: algebra and geometry. A Lie group is first of all a group. Secondly it is a smooth manifold which is a specific kind of geometric object. The circle and the sphere are examples of smooth manifolds. Finally the algebraic structure and the geometric structure must be compatible in a precise way.Informally, a Lie group is a group of symmetries where the symmetries are continuous. A circle has a continuous group of symmetries: you can rotate the circle an arbitrarily small amount and it looks the same. This is in contrast to the hexagon, for example. If you rotate the hexagon by a small amount then it will look different. Only rotations that are multiples of one-sixth of a full turn are symmetries of a hexagon.
Lie groups were studied by the Norwegian mathematician Sophus Lie at
the end of the 19 Lie groups are ubiquitous in mathematics and all areas of science. Associated to any system which has a continuous group of symmetries is a Lie group.
The basic building blocks of Lie groups are
Cartan constructed all the simple Lie algebras, which correspond to
the simple root systems:
A
For a Lie group, the subscript A, for example,
is certainly _{1000}larger than
E,
mathematicians know
how to describe the representations of
_{8}A for every _{n}n, so there is nothing special
about
A.
All of
_{1000}A,
_{n}B,
_{n}C, and
_{n}D are well-understood, so the remaining challenge
is to tackle the exceptional groups.
All 5 of the exceptional
groups need to be treated separately, and
_{n}E is the most complicated of these.
_{8} |