Intrinsic construction of invariant functions on simple Lie algebras

TL;DR

This paper introduces an intrinsic algorithm for constructing primitive invariant functions on complex simple Lie algebras, avoiding representations, with potential applications to exceptional Lie algebras.

Contribution

It presents a novel intrinsic method for constructing invariant functions on Lie algebras, especially useful for exceptional types, without relying on representations.

Findings

Algorithm successfully constructs primitive invariant functions

Implementation provided in Maple for future applications

Method applicable to complex simple Lie algebras, including exceptional types

Abstract

An algorithm for constructing primitive adjoint-invariant functions on a complex simple Lie algebra is presented. The construction is intrinsic in the sense that it does not resort to any representation. A primitive invariant function on the whole Lie algebra is obtained by lifting a coordinate function on a Kostant slice of the Lie algebra. Such an intrinsic construction of invariant functions is most useful for the bigger exceptional Lie algebras such as the E's. The Maple implementation of this algorithm is outlined at the end and will be applied to these exceptional Lie algebras in a future work.