A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable.
In a complex Lie group, the underlying smooth manifold is a complex manifold and the group operations are holomorphic maps. Forgetting the complex structure gives an ordinary Lie group whose real dimension is twice its complex dimension.
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 real numbers), 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, every positive-dimensional Lie group has a nonvanishing vector field. This structure has allowed complete classification of the finite dimensional semisimple Lie groups and their representations.
