Characteristic polynomials and finitely dimensional representations of simple Lie Algebras

TL;DR

This paper explores the relationship between finite dimensional representations of simple Lie algebras and their characteristic polynomials, introducing a monoid structure and analyzing specific cases like sl(2, C).

Contribution

It establishes a new correspondence between representations and characteristic polynomials and constructs a monoid structure on these polynomials.

Findings

Characteristic polynomials correspond to finite dimensional representations.

A monoid structure on characteristic polynomials is constructed.

Results on Borel and parabolic subalgebras via characteristic polynomials.

Abstract

In this paper, the correspondence between the finite dimensional representations of a simple Lie algebra and their characteristic polynomials is established, and a monoid structure on these characteristic polynomials is constructed. Furthermore, the characteristic polynomials of sl(2, C) on some classical simple Lie algebras through adjoint representations are studied, and we present some results of Borel subalgebras and parabolic subalgebras of simple Lie algebras through characteristic polynomials.