Editing Lie algebra representation (section)

===Homomorphisms===
Let <math>\mathfrak{g}</math> be a [[Lie algebra]]. Let ''V'', ''W'' be <math>\mathfrak{g}</math>-modules. Then a linear map <math>f: V \to W</math> is a '''homomorphism''' of <math>\mathfrak{g}</math>-modules if it is <math>\mathfrak{g}</math>-equivariant; i.e., <math>f(X\cdot v) = X\cdot f(v)</math> for any <math>X \in \mathfrak{g},\, v \in V</math>. If ''f'' is bijective, <math>V, W</math> are said to be '''equivalent'''. Such maps are also referred to as '''intertwining maps''' or '''morphisms'''.

Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.