Determinants and characteristic polynomials of Lie algebras

For an s-tuple A = (A(1), ... ,As) of square matrices of the same size, the (joint) determinant of A and the characteristic polynomial of A are defined by det(A) (z) = det(z(1)A(1) + z(2)A(2) + ... + z(s)A(s)) and p(A)(z) = det(z(0)I + z(1)A(1) + z(2)A(2) + ... + z(s)A(s)), respectively. This paper calculates determinant of the finite dimensional irreducible representations of s1(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. (C) 2018 Elsevier Inc. All rights reserved.