Editing Lie algebra representation (section)

==Formal definition==
Let <math>\mathfrak g</math> be a Lie algebra and let <math>V</math> be a vector space. We let <math>\mathfrak{gl}(V)</math> denote the space of endomorphisms of <math>V</math>, that is, the space of all linear maps of <math>V</math> to itself. Here, the associative algebra <math>\mathfrak{gl}(V)</math> is turned into a Lie algebra with bracket given by the commutator: <math>[s,t]=s \circ t-t \circ s</math> for all ''s,t'' in <math>\mathfrak{gl}(V)</math>. Then a '''representation''' of <math>\mathfrak g</math> on <math>V</math> is a [[Lie algebra homomorphism]]
:<math>\rho\colon \mathfrak g \to \mathfrak{gl}(V)</math>.
Explicitly, this means that <math>\rho</math> should be a linear map and it should satisfy
:<math>\rho([X,Y])=\rho(X)\rho(Y)-\rho(Y)\rho(X)</math>
for all ''X, Y'' in <math>\mathfrak g</math>. The vector space ''V'', together with the representation ''ρ'', is called a '''<math>\mathfrak g</math>-module'''.  (Many authors abuse terminology and refer to ''V'' itself as the representation).

The representation <math>\rho</math> is said to be '''faithful''' if it is injective.

One can equivalently define a <math>\mathfrak g</math>-module as a vector space ''V'' together with a [[bilinear map]] <math>\mathfrak g \times V\to V</math> such that
:<math>[X,Y]\cdot v = X\cdot(Y\cdot v) - Y\cdot(X\cdot v)</math>
for all ''X,Y'' in <math>\mathfrak g</math> and ''v'' in ''V''. This is related to the previous definition by setting ''X'' ⋅ ''v'' = ''ρ''(''X'')(''v'').