Determinants and characteristic polynomials of Lie algebras

Abstract For an s -tuple A = ( A 1 , … , A s ) 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 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.