A representation of a Lie algebra 
is a linear transformation
 |
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) |
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 irreducible representations of complex Lie algebras are determined by the classification of the semisimple
Lie algebras. Any irreducible representation 
of a complex Lie algebra 
is the tensor product 
,
where 
is an irreducible representation of
the quotient 
of the algebra 
and its Lie algebra radical,
and 
is a one-dimensional representation.
A Lie algebra may be associated with a Lie group, in which case it reflects the local structure of the Lie
group. Whenever a Lie group 
has a group representation
on 
,
its tangent space at the identity, which is a Lie algebra, has a Lie algebra
representation on 
given by the differential at the identity. Conversely, if
a connected Lie group 
corresponds to the Lie algebra 
, and 
has a Lie algebra representation
on 
,
then 
has a group representation on 
given by the matrix exponential.
