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
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.