The adjoint representation inside the exterior algebra of a simple Lie algebra

Abstract

For a simple complex Lie algebra $\mathfrak{g}$ we study the space of invariants $A={\left(\bigwedge {\mathfrak{g}}^{\ast }\otimes {\mathfrak{g}}^{\ast }\right)}^{\mathfrak{g}}$, (which describes the isotypic component of type $\mathfrak{g}$ in $\bigwedge {\mathfrak{g}}^{\ast }$) as a module over the algebra of invariants ${\left(\bigwedge {\mathfrak{g}}^{\ast }\right)}^{\mathfrak{g}}$. 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 $(\bigwedge {\mathfrak{g}}^{\ast }{)}^{\mathfrak{g}}$, with the exception of the one of highest degree.

Final version. More misprints corrected. To appear in Advances in Mathematics

References in corpus (3)

• On the theorem of Amitsur--Levitzki
• On certain modules of covariants in exterior algebras
• Quasi-identities on matrices and the Cayley-Hamilton polynomial

Cited by in corpus (4)

• The images of Lie polynomials evaluated on matrices
• On certain modules of covariants in exterior algebras
• Quasi-identities on matrices and the Cayley-Hamilton polynomial
• On special covariants in the exterior algebra of a simple Lie algebra