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

In this paper, we obtain a general formula for the characteristic polynomial of a finitely dimensional representation of Lie algebra $\mathfrak{s}\mathfrak{l}(2,\text{C})$ 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.