# Adjoint representation of a Lie algebra of a same dimension that the basis representa

 Sci Advisor HW Helper P: 4,300 I'm not very well-founded in Lie-algebra's, but the adjoint of an element x is the map $$\operatorname{ad}_x: y \mapsto [x, y]$$ isn't it? So the adjoint is linear, i.e. $$\operatorname{ad}_x + \operatorname{ad}_y = \operatorname{ad}_{x + y}, \operatorname{ad}_x(y) + \operatorname{ad}_x(z) = \operatorname{ad}(y + z)$$ etc. - then isn't the adjoint representation always of the same dimension. I.e. the basic representation provides the generators $g_i$ of the Lie-algebra, and then $\operatorname{ad}_{g_i}$ generate the adjoint representation?