In this paper, we prove the correspondence between finite dimensional representations of a simple Lie algebra and their associated characteristic polynomials. We will also define a monoid structure on these characteristic polynomials related to the tensor products of the representations. Furthermore, the factorization of characteristic polynomials sheds new light on the structure of simple Lie algebras and their Borel subalgebras.

Liu is funded by the NSFC (Grant No. 11971181, Grant No.12271298).