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The adjoint representation inside the exterior algebra of a simple Lie algebra. (English) Zbl 1370.17014
Summary: For a simple complex Lie algebra $$\mathfrak{g}$$ we study the space of invariants $$A = (\land \mathfrak{g}^\ast \otimes \mathfrak{g}^\ast)^{\mathfrak{g}}$$, which describes the isotypic component of type $$\mathfrak{g}$$ in $$\land \mathfrak{g}^\ast$$, as a module over the algebra of invariants $$(\land \mathfrak{g}^\ast)^{\mathfrak{g}}$$. As main result we prove that $$A$$ is a free module, of rank twice the rank of $$\mathfrak{g}$$, over the exterior algebra generated by all primitive invariants in $$(\land \mathfrak{g}^\ast)^{\mathfrak{g}}$$, with the exception of the one of highest degree.

##### MSC:
 17B20 Simple, semisimple, reductive (super)algebras
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##### References:
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