Sufficient Condition for Adjoint Representation Kernel to be Lie Group Center

[kakarukeys]

what is the sufficient condition for the kernel of an adjoint representation to be the center of the Lie group?

Does the Lie group have to be non-compact and connected, etc?

 

[LorenzoMath]

If the Lie group is connected, the claim holds. So I heard. Is there any chance that you are studying automorphic representations and their L-functions? If so, please let me know good references to start with. Also, references for representation theory. Thanks.

 

[tacman]

The sufficient condition for the kernel of an adjoint representation to be the center of the Lie group is that the Lie group must be compact and connected. This means that the group is a closed and connected set, and all its elements can be continuously connected to the identity element. Additionally, the Lie group must also have a trivial center, meaning that the only element in the center is the identity element.

If these conditions are met, then the kernel of the adjoint representation will be the center of the Lie group. This is because the kernel of the adjoint representation consists of all elements that commute with all other elements in the Lie group. Therefore, if the center of the Lie group is trivial, the only element that satisfies this condition is the identity element, making it the kernel of the adjoint representation.

It is important to note that this is not the only sufficient condition for the kernel of the adjoint representation to be the center of the Lie group. Other conditions may exist, but the compactness and connectedness of the Lie group are commonly used and well-known conditions.

In summary, the sufficient condition for the kernel of an adjoint representation to be the center of the Lie group is that the group must be compact, connected, and have a trivial center.