The Classical Family Algebra of the Adjoint Representation of $sl(n)$

Abstract

For the simple Lie algebra $g = sl(n,C)$ we we find a set of generators and relations for the classical family algebra $(End(g)\otimes S(g))^G$ as an algebra over the ring $I(g)$. From these we can then determine a $I(g)$-linear basis of the family algebra, and thus the generalized exponents of the irreducible components of $End(g)$ viewed as a $g$-module.