# For an \$n\$-dimensional Lie algebra, is there always a matrix representation \$\{M\}\$ and a single vector \$v\$ such that \$\{Mv\}\$ is \$n\$-dimensional?

by mr_e_man   Last Updated September 11, 2019 18:20 PM - source

For an $$n$$-dimensional Lie algebra, is there always a matrix representation $$\{M\}$$ and a single vector $$v$$ such that $$\{Mv\}$$ is $$n$$-dimensional?

This would necessarily be a faithful representation.

I'm focusing on Lie algebras over $$\mathbb R$$, but more general answers are welcome.

This might have something to do with weights or Whitehead's lemma, but I don't know enough about representation theory to be sure.

Tags :