A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable.
This definition is related to the fifth of Hilbert's problems, which asks if the assumption of differentiability for functions defining a continuous transformation group can be avoided.
The simplest examples of Lie groups are one-dimensional. Under addition, the real line is a Lie group. After picking a specific point to be the identity element, the circle is also a Lie group. Another point on the circle at angle from the identity then acts by rotating the circle by the angle In general, a Lie group may have a more complicated group structure, such as the orthogonal group (i.e., the orthogonal matrices), or the general linear group (i.e., the invertible matrices). The Lorentz group is also a Lie group.
The tangent space at the identity of a Lie group always has the structure of a Lie algebra, and this Lie algebra determines the local structure of the Lie group via the exponential map. For example, the function gives the exponential map from the circle's tangent space (i.e., the reals), to the circle, thought of as a unit circle in . A more difficult example is the exponential map from antisymmetric matrices to the special orthogonal group , the subset of with determinant 1.
The topology of a Lie group is fairly restricted. For example, there always exists a nonvanishing vector field. This structure has allowed complete classification of the finite dimensional semisimple Lie groups and their representations.