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

TL;DR

This paper develops a detailed algebraic structure for the classical family algebra associated with the adjoint representation of sl(n), providing generators, relations, and basis to understand its composition.

Contribution

It introduces a set of generators and relations for the classical family algebra of sl(n), enabling explicit basis construction and analysis of irreducible components.

Findings

Derived generators and relations for the family algebra.

Established a basis over the ring I(g).

Determined generalized exponents of irreducible components.

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.