Editing Adjoint representation (section)

=== Example SL(2, R) ===
When computing the root system for one of the simplest cases of Lie Groups, the group SL(2, '''R''') of two dimensional matrices with determinant 1 consists of the set of matrices of the form:

: <math>\begin{bmatrix}
a & b\\
c & d\\
\end{bmatrix} </math>

with ''a'', ''b'', ''c'', ''d'' real and ''ad''&nbsp;&minus;&nbsp;''bc''&nbsp;=&nbsp;1.

A maximal compact connected abelian Lie subgroup, or maximal torus ''T'', is given by the subset of all matrices of the form

: <math>\begin{bmatrix}
t_1 & 0\\
0 & t_2\\
\end{bmatrix}
=
\begin{bmatrix}
t_1 & 0\\
0 & 1/t_1\\
\end{bmatrix}
=
\begin{bmatrix}
\exp(\theta) & 0 \\
0 & \exp(-\theta) \\
\end{bmatrix} </math>

with <math> t_1 t_2 = 1 </math>. The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices

: <math>
\begin{bmatrix}
\theta & 0\\
0 & -\theta \\
\end{bmatrix} = 
\theta\begin{bmatrix}
1 & 0\\
0 & 0 \\
\end{bmatrix}-\theta\begin{bmatrix}
0 & 0\\
0 & 1 \\
\end{bmatrix}
= \theta(e_1-e_2).
</math>

If we conjugate an element of SL(2, ''R'') by an element of the maximal torus we obtain

: <math>
\begin{bmatrix}
t_1 & 0\\
0 & 1/t_1\\
\end{bmatrix}
\begin{bmatrix}
a & b\\
c & d\\
\end{bmatrix}
\begin{bmatrix}
1/t_1 & 0\\
0 & t_1\\
\end{bmatrix}
=
\begin{bmatrix}
a t_1 & b t_1 \\
c / t_1 & d / t_1\\
\end{bmatrix}
\begin{bmatrix}
1 / t_1 & 0\\
0       & t_1\\
\end{bmatrix}
=
\begin{bmatrix}
a       & b t_1^2\\
c t_1^{-2} & d\\
\end{bmatrix}
</math>

The matrices

: <math>
\begin{bmatrix}
1 & 0\\
0 & 0\\
\end{bmatrix}
\begin{bmatrix}
0 & 0\\
0 & 1\\
\end{bmatrix}
\begin{bmatrix}
0 & 1\\
0 & 0\\
\end{bmatrix}
\begin{bmatrix}
0 & 0\\
1 & 0\\
\end{bmatrix}
</math>

are then 'eigenvectors' of the conjugation operation with eigenvalues <math>1,1,t_1^2, t_1^{-2}</math>. The function Λ which gives <math>t_1^2</math> is a multiplicative character, or homomorphism from the group's torus to the underlying field R. The function λ giving θ is a weight of the Lie Algebra with weight space given by the span of the matrices.

It is satisfying to show the multiplicativity of the character and the linearity of the weight. It can further be proved that the differential of Λ can be used to create a weight. It is also educational to consider the case of SL(3, '''R''').