- #1
AlbertEi
- 27
- 0
Hi,
I'm getting a bit confused about the adjoint representation. I learned about Lie algrebras using the book by Howard Georgi (i.e. it is very "physics-like" and we did not distinguish between the abstract approach to group theory and the matrix approach to group theory). He defines the adjoint as follows:
\begin{equation}
[T_a]_{bc} = i f_{abc} T_c
\end{equation}
where $T_i$ are the generators and $f_abc$ are the structure constants. This definition clearly means that matrix in the adjoint representation must have the same amount of dimensions as there are generators. However, recently I have reading some papers where they talk about the adjoint representation of for instance the Higgs field $\phi$ as follows:
\begin{equation}
\phi=\phi^a T^a
\end{equation}
with the following transformation properties:
\begin{equation}
\phi \mapsto g \phi g^{-1}
\end{equation}
where the generators do not necessarily have to be of the same dimensions as the number of generators. I understand that this basically just means that $\phi$ always takes values in the Lie algebra, but I think it is really odd that two very different properties of the Lie algebra have been given the same name. My confusion even grows further when they discuss roots/weight. More specifically, are the eigenvalues of the latter generators called roots or weights?
I'm getting a bit confused about the adjoint representation. I learned about Lie algrebras using the book by Howard Georgi (i.e. it is very "physics-like" and we did not distinguish between the abstract approach to group theory and the matrix approach to group theory). He defines the adjoint as follows:
\begin{equation}
[T_a]_{bc} = i f_{abc} T_c
\end{equation}
where $T_i$ are the generators and $f_abc$ are the structure constants. This definition clearly means that matrix in the adjoint representation must have the same amount of dimensions as there are generators. However, recently I have reading some papers where they talk about the adjoint representation of for instance the Higgs field $\phi$ as follows:
\begin{equation}
\phi=\phi^a T^a
\end{equation}
with the following transformation properties:
\begin{equation}
\phi \mapsto g \phi g^{-1}
\end{equation}
where the generators do not necessarily have to be of the same dimensions as the number of generators. I understand that this basically just means that $\phi$ always takes values in the Lie algebra, but I think it is really odd that two very different properties of the Lie algebra have been given the same name. My confusion even grows further when they discuss roots/weight. More specifically, are the eigenvalues of the latter generators called roots or weights?
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