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J Lodder
J Lodder
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When is there a g$g$-module isomorphism between a semi-simple Lie algebra g$g$ and an exterior power of its standard representation?

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Harry Gindi
Harry Gindi
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When is there a $g$g-module isomorphism between a semi-simple Lie algebra $g$g and an exterior power of its standard representation?

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J Lodder
J Lodder
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When is there a $g$-module isomorphism between a semi-simple Lie algebra $g$ and an exterior power of its standard representation?

Let $g$ be a semi-simple Lie algebra, and $0 \to I \to h \to g \to 0$ an Abelian extension of $g$. Then $g$ acts on $I$. Considering $g$ under the adjoint action, when is there a $g$-module isomorphism between $g$ and the k-th exterior power $\Lambda^k(I)$ for some $k$? Only when $g = so(n)$, $k=2$, $k = n-2$?