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Representation Lie Algebra
A representation of a Lie algebra
is a Lie algebra homomorphism
is the commutator Lie
algebra
of some vector space
. In other words, 
is a linear
mapping
that satisfies
![$\displaystyle \rho([a,b]) = \rho(a)\rho(b)-\rho(b)\rho(a),\quad a,b\in\mathfrak{g}$](http://myyn.org/static/assets/mathbrowser/article/8228/images/img6.png){kind=small}
a
-module, and calls
the action
of 
on 
.
We call the representation faithful if 
is injective.
A invariant subspace
or sub-module
is a subspace
of 
satisfying
for all
. A representation is
called irreducible or simple
if its only invariant subspaces are 
and the whole representation.
The dimension of 
is called the dimension of the representation.
If 
is infinite-dimensional, then one speaks of an
infinite-dimensional representation.
Given a pair of representations, we can define a new representation, called the direct sum of the two given representations:
If
and
are representations, then 
has the obvious
Lie algebra action, by the embedding
.