Editing Lie algebra representation (section)

==Basic constructions==
===Tensor products of representations===
{{main|Tensor product of representations}}
If we have two representations of a Lie algebra <math>\mathfrak{g}</math>, with ''V''<sub>1</sub> and ''V''<sub>2</sub> as their underlying vector spaces, then the tensor product of the representations would have ''V''<sub>1</sub> ⊗ ''V''<sub>2</sub> as the underlying vector space, with the action of <math>\mathfrak{g}</math> uniquely determined by the assumption that

:<math>X\cdot(v_1\otimes v_2)=(X\cdot v_1)\otimes v_2+v_1\otimes (X\cdot v_2) .</math>
for all <math>v_1\in V_1</math> and <math>v_2\in V_2</math>.

In the language of homomorphisms, this means that we define <math>\rho_1\otimes\rho_2:\mathfrak{g}\rightarrow\mathfrak{gl}(V_1\otimes V_2) </math> by the formula 
:<math>(\rho_1\otimes\rho_2)(X)=\rho_1(X)\otimes \mathrm{I}+\mathrm{I}\otimes\rho_2(X)</math>.<ref>{{harvnb|Hall|2015}} Section 4.3</ref> This is called the Kronecker sum of <math>\rho_1</math> and <math>\rho_2</math>, defined in [[Matrix addition#Kronecker_sum]] and [[Kronecker product#Properties]], and more specifically in [[Tensor product of representations]]. 

In the physics literature, the tensor product with the identity operator is often suppressed in the notation, with the formula written as
:<math>(\rho_1\otimes\rho_2)(X)=\rho_1(X)+\rho_2(X)</math>,
where it is understood that <math>\rho_1(x)</math> acts on the first factor in the tensor product and <math>\rho_2(x)</math> acts on the second factor in the tensor product. In the context of representations of the Lie algebra su(2), the tensor product of representations goes under the name "addition of angular momentum." In this context, <math>\rho_1(X)</math> might, for example, be the orbital angular momentum while <math>\rho_2(X)</math> is the spin angular momentum.

===Dual representations===
{{main|Dual representation}}
Let <math>\mathfrak{g}</math> be a Lie algebra and <math>\rho:\mathfrak{g}\rightarrow\mathfrak{gl}(V)</math> be a representation of <math>\mathfrak{g}</math>. Let <math>V^*</math> be the dual space, that is, the space of linear functionals on <math>V</math>. Then we can define a representation <math>\rho^*:\mathfrak{g}\rightarrow\mathfrak{gl}(V^*)</math> by the formula
:<math>\rho^*(X)=-(\rho(X))^\operatorname{tr},</math>
where for any operator <math>A:V\rightarrow V</math>, the transpose operator <math>A^\operatorname{tr}:V^*\rightarrow V^*</math> is defined as the "composition with <math>A</math>" operator:
:<math>(A^\operatorname{tr}\phi)(v)=\phi(Av)</math>
The minus sign in the definition of <math>\rho^*</math> is needed to ensure that <math>\rho^*</math> is actually a representation of <math>\mathfrak{g}</math>, in light of the identity <math>(AB)^\operatorname{tr}=B^\operatorname{tr}A^\operatorname{tr}.</math>

If we work in a basis, then the transpose in the above definition can be interpreted as the ordinary matrix transpose.

===Representation on linear maps===

Let <math>V, W</math> be <math>\mathfrak{g}</math>-modules, <math>\mathfrak{g}</math> a Lie algebra. Then <math>\operatorname{Hom}(V, W)</math> becomes a <math>\mathfrak{g}</math>-module by setting <math>(X \cdot f)(v) = X f(v) - f (X v)</math>. In particular, <math>\operatorname{Hom}_\mathfrak{g}(V, W) = \operatorname{Hom}(V, W)^\mathfrak{g}</math>; that is to say, the <math>\mathfrak{g}</math>-module homomorphisms from <math>V</math> to <math>W</math> are simply the elements of <math>\operatorname{Hom}(V, W)</math> that are invariant under the just-defined action of <math>\mathfrak{g}</math> on <math>\operatorname{Hom}(V, W)</math>. If we take <math>W</math> to be the base field, we recover the action of <math>\mathfrak{g}</math> on <math>V^*</math> given in the previous subsection.