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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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