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.
