Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If two finite dimensional matrix Lie algebras are isomorphic, is it always possible to see the isomorphism as a similarity transformation $g \mapsto M^{-1} g M$ ?

share|improve this question
    
Are you asking: given two subalgebras $g$ and $g'$ of the matrx algebra $M_n(k)$, is there a matrix $M$ such that $g'=MgM^{-1}$? –  Mariano Suárez-Alvarez Mar 27 '13 at 19:19
add comment

1 Answer

If the question is

given two isomorphic Lie subalgebras $\mathfrak g$, $\mathfrak g'$ of $M_n(k)$, does there exist a matrix $M$ such that $\mathfrak g'=M\mathfrak gM^{-1}$?

then the answer is no. Just take $\mathfrak g$ and $\mathfrak g'$ to be two $1$-dimensional Lie algebras spanned by matrices which are not conjugated.

share|improve this answer
    
In fact, in some sense, the whole theory of representations of Lie algebras gives a ton of examples: Any two non-equivalent faithful representations of the same dimension give a counter example. –  Jason DeVito Mar 27 '13 at 19:22
    
Is it possible that two non-equivalent faithful representations of the same dimension that are as well irreducible are non equivalent? Is there probably a difference between semisimple or general Lie algebras? –  user9784 Mar 31 '13 at 17:19
    
@user9784 I don't understand the question. –  Mariano Suárez-Alvarez Mar 31 '13 at 21:11
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.