Irreducible representations of simple Lie algebras by differential operators

TL;DR

This paper introduces a systematic method to construct all highest-weight modules of simple Lie algebras using differential operators, providing explicit formulas for classical cases and highlighting the universality of raising operators.

Contribution

It presents a novel, universal differential operator framework for constructing finite-dimensional irreducible representations of simple Lie algebras, including explicit formulas for classical types.

Findings

Constructed highest-weight modules via differential operators in a systematic way.

Provided explicit formulas for simple root generators of classical Lie algebras.

Demonstrated the universality of raising operators across representations.

Abstract

We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $\mathfrak{g}$. The Lie algebra generators are represented as first order differential operators in $\frac{1}{2} \left(\dim \mathfrak{g} - \text{rank} \, \mathfrak{g}\right) $ variables. All rising generators ${\bf e}$ are universal in the sense that they do not depend on representation, the weights enter (in a very simple way) only in the expressions for the lowering operators ${\bf f}$. We present explicit formulas of this kind for the simple root generators of all classical Lie algebras.