# Lie group

Part of the E_{8} matrix.
Credit: David Vogan, MIT .

A Lie group is a group that is also a manifold. Lie groups were first described in the nineteenth century by the Norwegian mathematician Sophus Lie (pronounced 'Lee'). Lie groups of real matrices, such as occur in quantum field theory, give naturally occurring examples of Lie groups. The tangent space at the identity element of a Lie group forms a Lie algebra in a natural way.

In layperson's terms, a Lie group is a way of describing symmetrical objects.
For example, there are Lie groups to describe the symmetry of simple objects
such as balls, cylinders, cones. The most complicated Lie group that exists
and, therefore, the most complicated symmetry known is a 248-dimensional
structure called E_{8}.

## E_{8}

The Lie group known as E_{8} was discovered in 1887 but fully mapped
only in 2007 after several years of effort by a team of 18 mathematicians.
The final expression of it consisted of a 453,060 × 453,060 matrix.
Each of the 205,263,363,600 entries in this matrix consists of equations,
some of which are quite intricate.

E_{8} is much more than a mathematical curiosity, however. Physicists
have encountered it increasingly in their efforts to unify gravity with
the other fundamental forces in nature through theories such as string
theory. The mapping of E_{8} may thus play a significant role
in developing a consistent theory of quantum gravity.