Invariance of Commutator Relations

[fresh_42]

Does anybody know of examples, in which groups defined by $[\varphi(X),\varphi(Y)]=[X,Y]$ are investigated? The $X,Y$ are vectors of a Lie algebra, so imagine them to be differential operators, or vector fields, or as physicists tend to say: generators. The $\varphi$ are thus linear transformations of named Lie algebra that preserve the commutation relations.

 

[martinbn]

So, you are asking about groups of automorphisms of Lie algebras?

 

[fresh_42]

So, you are asking about groups of automorphisms of Lie algebras?

No. That would be $[\varphi(X),\varphi(Y)]=\varphi([X,Y])$. I asked about $\ldots = [X,Y]\,.$

In other words: a homomorphism is of the form $\varphi^* \otimes \varphi^* \otimes \varphi^{-1}=\operatorname{id}$ and I am interested in $\varphi^* \otimes \varphi^* \otimes 1=\operatorname{id}\,.$