  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.
