Determinants and characteristic polynomials of Lie algebras

For an s-tuple A=(A1,…,As) of square matrices of the same size, the (joint) determinant of A and the characteristic polynomial of A are defined bydet⁡(A)(z)=det⁡(z1A1+z2A2+⋯+zsAs) andpA(z)=det⁡(z0I+z1A1+z2A2+⋯+zsAs), respectively. This paper calculates determinant of the finite dimensional irreducible representations of sl(2,F), which is either zero or a product of some irreducible quadratic polynomials. Moreover, it shows that a finite dimensional Lie algebra is solvable if and only if the characteristic polynomial is completely reducible.