maverick280857
- 1,774
- 5
Hi,
I am trying to work through a proof/argument to show that the adjoint representation of a semisimple Lie algebra is completely reducible.
Suppose S denotes an invariant subspace of the Lie algebra, and we pick Y_i in the invariant subspace S. The rest of the generators X_r are such that the natural inner product is (X_r, Y_i) = 0. This can be done by some suitable Gram Schmidt orthogonalization if necessary.
To begin with, I argue that the killing metric in this basis is block diagonal. If i denotes an index on Y and r denotes an index on X, then g_{ir} = 0 as the Killing form is the natural inner product or its negative depending on whether Y is chosen to be symmetric or antisymmetric. This is OK.
But the following argument is unclear to me
Since S is an invariant subspace, structure constants of the form {f_{ir}}^{s} are zero.
Is it reasonable to expect [X, Y] to be in S as well as its complement? The only way then that this would be possible is if [X, Y] = 0.
The other argument (specious to me) is that Y and X live in different spaces so they must commute. This seems physically reasonable, but I don't see how to argue this mathematically.
Any help would be greatly appreciated. Oh and I should point out, I am learning this from the standpoint of a theoretical physicist, so please feel free to point out mistakes/improvements in the reasoning (or holes in my understanding) from a purely mathematical perspective.
Thanks!
I am trying to work through a proof/argument to show that the adjoint representation of a semisimple Lie algebra is completely reducible.
Suppose S denotes an invariant subspace of the Lie algebra, and we pick Y_i in the invariant subspace S. The rest of the generators X_r are such that the natural inner product is (X_r, Y_i) = 0. This can be done by some suitable Gram Schmidt orthogonalization if necessary.
To begin with, I argue that the killing metric in this basis is block diagonal. If i denotes an index on Y and r denotes an index on X, then g_{ir} = 0 as the Killing form is the natural inner product or its negative depending on whether Y is chosen to be symmetric or antisymmetric. This is OK.
But the following argument is unclear to me
Since S is an invariant subspace, structure constants of the form {f_{ir}}^{s} are zero.
Is it reasonable to expect [X, Y] to be in S as well as its complement? The only way then that this would be possible is if [X, Y] = 0.
The other argument (specious to me) is that Y and X live in different spaces so they must commute. This seems physically reasonable, but I don't see how to argue this mathematically.
Any help would be greatly appreciated. Oh and I should point out, I am learning this from the standpoint of a theoretical physicist, so please feel free to point out mistakes/improvements in the reasoning (or holes in my understanding) from a purely mathematical perspective.
Thanks!
Last edited: