Characteristic polynomials and finitely dimensional representations of $\mathfrak{sl}(2, \mathbb{C})$

TL;DR

This paper derives a general formula for the characteristic polynomial of finite-dimensional representations of rak{sl}(2, \u00C4) and establishes a one-to-one correspondence between representations and these polynomials, introducing a monoid structure.

Contribution

It provides a universal formula for characteristic polynomials of rak{sl}(2, \u00C4) representations and links them to a monoid structure, revealing a new algebraic perspective.

Findings

One-to-one correspondence between representations and characteristic polynomials.

A new monoid structure on characteristic polynomials.

Explicit formula for characteristic polynomials of rak{sl}(2, \u00C4) representations.

Abstract

In this paper, we obtain a general formula for the characteristic polynomial of a finitely dimensional representation of Lie algebra $\mathfrak{sl}(2, \C )$ and the form for these characteristic polynomials, and prove there is one to one correspondence between representations and their characteristic polynomials. We define a product on these characteristic polynomials, endowing them with a monoid structure.