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added a characteristic condition
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Tushant Mittal
Tushant Mittal
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For a simple lie algebra  $\mathfrak{g}$ over a field of characteristic 0, define $\mathfrak{o}(k)$ to be the orthogonal lie algebra with respect to the Killing form.

In the proof of Theorem 2 in the following paper, https://arxiv.org/pdf/math/0407240.pdf the author mentions that the following is true,

The $\mathfrak{g}$ module $\mathfrak{o}(k)/ad(\mathfrak{g})$ is irreducible if $\mathfrak{g}$ is not of type A while it is a direct sum $W \oplus W^∗ $for some non-self-dual module $W$ if $\mathfrak{g}$ is of type A.

I am unable to see why this is true. Can someone furnish a proof or point to references?

I've also asked it here - https://math.stackexchange.com/questions/3335347/irreducibility-of-the-mathfrakg-module-mathfrakok-ad-mathfrakg

For a simple lie algebra  $\mathfrak{g}$, define $\mathfrak{o}(k)$ to be the orthogonal lie algebra with respect to the Killing form.

In the proof of Theorem 2 in the following paper, https://arxiv.org/pdf/math/0407240.pdf the author mentions that the following is true,

The $\mathfrak{g}$ module $\mathfrak{o}(k)/ad(\mathfrak{g})$ is irreducible if $\mathfrak{g}$ is not of type A while it is a direct sum $W \oplus W^∗ $for some non-self-dual module $W$ if $\mathfrak{g}$ is of type A.

I am unable to see why this is true. Can someone furnish a proof or point to references?

I've also asked it here - https://math.stackexchange.com/questions/3335347/irreducibility-of-the-mathfrakg-module-mathfrakok-ad-mathfrakg

For a simple lie algebra $\mathfrak{g}$ over a field of characteristic 0, define $\mathfrak{o}(k)$ to be the orthogonal lie algebra with respect to the Killing form.

In the proof of Theorem 2 in the following paper, https://arxiv.org/pdf/math/0407240.pdf the author mentions that the following is true,

The $\mathfrak{g}$ module $\mathfrak{o}(k)/ad(\mathfrak{g})$ is irreducible if $\mathfrak{g}$ is not of type A while it is a direct sum $W \oplus W^∗ $for some non-self-dual module $W$ if $\mathfrak{g}$ is of type A.

I am unable to see why this is true. Can someone furnish a proof or point to references?

I've also asked it here - https://math.stackexchange.com/questions/3335347/irreducibility-of-the-mathfrakg-module-mathfrakok-ad-mathfrakg

Source Link
Tushant Mittal
Tushant Mittal
  • 143
  • 4

Irreducibility of the $\mathfrak{g}$-module $\mathfrak{o}(k)/ad(\mathfrak{g})$

For a simple lie algebra $\mathfrak{g}$, define $\mathfrak{o}(k)$ to be the orthogonal lie algebra with respect to the Killing form.

In the proof of Theorem 2 in the following paper, https://arxiv.org/pdf/math/0407240.pdf the author mentions that the following is true,

The $\mathfrak{g}$ module $\mathfrak{o}(k)/ad(\mathfrak{g})$ is irreducible if $\mathfrak{g}$ is not of type A while it is a direct sum $W \oplus W^∗ $for some non-self-dual module $W$ if $\mathfrak{g}$ is of type A.

I am unable to see why this is true. Can someone furnish a proof or point to references?

I've also asked it here - https://math.stackexchange.com/questions/3335347/irreducibility-of-the-mathfrakg-module-mathfrakok-ad-mathfrakg