A Lie algebra representation of a Lie algebra 
is a linear transformation
 |
(1)
|
where 
is the set of all linear
transformations of a vector space 
. In particular, if 
, then 
is the set of 
square matrices. The
map 
is required to be a map of Lie
algebras so that
![psi([A,B])=psi(A)psi(B)-psi(B)psi(A)](/images/equations/LieAlgebraRepresentation/NumberedEquation2.svg) |
(2)
|
for all 
. Note that the expression 
only makes sense as a matrix
product in a representation. For example, if 
and 
are antisymmetric matrices,
then 
is skew-symmetric, but 
may not be antisymmetric.
The possible finite-dimensional irreducible representations of finite-dimensional
complex Lie algebras are determined by the classification
of the semisimple Lie algebras. Any finite-dimensional irreducible representation 
of a complex Lie algebra 
is the vector space tensor product 
, where 
is an irreducible representation of the quotient 
of the Lie algebra 
and its Lie algebra radical (the largest solvable subspace invariant under the adjoint
representation), and 
is a one-dimensional Lie algebra
representation.
A Lie algebra may be associated with a Lie group, in which case it reflects the local structure of the Lie
group. A smooth group representation 
of a Lie group 
on a finite-dimensional vector space 
induces a Lie algebra representation 
of its Lie algebra 
by taking the differential
at the identity element. Conversely, if 
is connected and simply
connected, any Lie algebra representation 
of 
on a finite-dimensional vector space 
integrates uniquely to a smooth group
representation 
of 
.
These representations are related by the
exponential map of 
and the matrix exponential:
 |
(3)
|
