Editing Lie algebra representation (section)

===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.