Editing Semisimple Lie algebra (section)

== Example root space decomposition in sl<sub>n</sub>(C)  ==
For <math>\mathfrak{g} = \mathfrak{sl}_n(\mathbb{C})
</math> and the Cartan subalgebra <math>\mathfrak{h}</math> of diagonal matrices, define <math>\lambda_i \in \mathfrak{h}^*</math> by
:<math>\lambda_i(d(a_1,\ldots, a_n)) = a_i</math>,
where <math>d(a_1,\ldots, a_n)</math> denotes the diagonal matrix with <math>a_1,\ldots, a_n</math> on the diagonal. Then the decomposition is given by
:<math>\mathfrak{g} = \mathfrak{h}\oplus \left( \bigoplus_{i \neq j} \mathfrak{g}_{\lambda_i - \lambda_j} \right)</math>
where
:<math>\mathfrak{g}_{\lambda_i - \lambda_j} = \text{Span}_\mathbb{C}(e_{ij})</math>
for the vector <math>e_{ij}</math> in <math>\mathfrak{sl}_n(\mathbb{C})</math> with the standard (matrix) basis, meaning <math>e_{ij}</math> represents the basis vector in the <math>i</math>-th row and <math>j</math>-th column. This decomposition of <math>\mathfrak{g}</math> has an associated root system:
:<math>\Phi = \{ \lambda_i - \lambda_j : i \neq j \}</math>

=== sl<sub>2</sub>(C) ===
For example, in <math>\mathfrak{sl}_2(\mathbb{C})</math> the decomposition is
:<math>\mathfrak{sl}_2= \mathfrak{h}\oplus
\mathfrak{g}_{\lambda_1 - \lambda_2}\oplus
\mathfrak{g}_{\lambda_2 - \lambda_1}</math>
and the associated root system is
:<math>\Phi = \{\lambda_1 - \lambda_2, \lambda_2 - \lambda_1 \}</math>

=== sl<sub>3</sub>(C) ===
In <math>\mathfrak{sl}_3(\mathbb{C})</math> the decomposition is
:<math>\mathfrak{sl}_3 = \mathfrak{h} \oplus \mathfrak{g}_{\lambda_1 - \lambda_2}
\oplus \mathfrak{g}_{\lambda_1 - \lambda_3}
\oplus \mathfrak{g}_{\lambda_2 - \lambda_3}

\oplus \mathfrak{g}_{\lambda_2 - \lambda_1}
\oplus \mathfrak{g}_{\lambda_3 - \lambda_1}
\oplus \mathfrak{g}_{\lambda_3 - \lambda_2}
</math>
and the associated root system is given by
:<math>\Phi = \{\pm(\lambda_1 - \lambda_2),\pm(\lambda_1 - \lambda_3),\pm(\lambda_2 - \lambda_3) \}</math>