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

Hello,

I hope it's not the wrong forum for my question which is the following:

Is there some list of Lie algebras, whose adjoint representations have the same dimension as their basic representation (like, e.g., this is the case for so(3))? How can one find such Lie algebras? Could you recommend some literature to me?

 Blog Entries: 5 Recognitions: Homework Help Science Advisor 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?