Series of Lie groups Academic Article uri icon

abstract

  • For various series of complex semi-simple Lie algebras $fg (t)$ equipped with irreducible representations $V(t)$, we decompose the tensor powers of $V(t)$ into irreducible factors in a uniform manner, using a tool we call {it diagram induction}. In particular, we interpret the decompostion formulas of Deligne cite{del} and Vogel cite{vog} for decomposing $fg^{ot k}$ respectively for the exceptional series and $kleq 4$ and all simple Lie algebras and $kleq 3$, as well as new formulas for the other rows of Freudenthal's magic chart. By working with Lie algebras augmented by the symmetry group of a marked Dynkin diagram, we are able to extend the list cite{brion} of modules for which the algebra of invariant regular functions under a maximal nilpotent subalgebra is a polynomial algebra. Diagram induction applied to the exterior algebra furnishes new examples of distinct representations having the same Casimir eigenvalue.

author list (cited authors)

  • Landsberg, J. M., & Manivel, L.

citation count

  • 17

publication date

  • August 2004