How do you tell if lie groups are isomorphic

[geoduck]

How can you tell if two Lie groups are isomorphic to each other?

If you have a set of generators, Tⁱ, then you can perform a linear transformation:

T'ⁱ=a_ijT^j

and these new generators T' will have different structure constants than T.

Isn't it possible to always find a linear transformation a_ij to make the structure constants anything you want?

 

[fzero]

The structure constants define the Lie algebra, not the group itself. Structure constants are, as you note, defined with respect to a particular basis. However, the change of basis corresponds to the action of the group on its algebra.

For example, the elements of the Lie algebras of the unitary groups are Hermitian matrices. A change of basis is a unitary transformation, which is an element of the corresponding unitary group. These transformations are called (inner) automorphisms of the Lie algebra. Generators which are related by a unitary transformation define the same Lie algebra.

Isomorphisms between groups are invertible group homomorphisms. In order to recognize the existence of isomorphisms, you want to examine the structure of the algebras more closely. The details of the root system and corresponding Dynkin diagram can be used to classify semisimple Lie algebras: http://en.wikipedia.org/wiki/Semisimple_Lie_algebra#Classification. Isomorphisms between these Lie algebras correspond to geometrical isomorphisms between their Dynkin diagrams.