Characteristic identities for semi-simple Lie algebras

Abstract

AbstractWe present a new derivation of the polynomial identities satisfied by certain matrices A with entries Aij (i, j = 1,…, n) from the universal enveloping algebra of a semi-simple Lie algebra. These polynomial identities are exhibited in a representation-independent way as p(A) = 0 where p(x) (herein called the characteristic polynomial of A) is a polynomial with coefficients from the centre Z of the universal enveloping algebra. The minimum polynomial identity m(A) = 0 of the matrix A over Z is also obtained and it is shown that p(x) and m(x) possess properties analogous to the characteristic and minimum polynomials respectively of a matrix with numerical entries. Acting on a representation (finite or infinite dimensional) admitting an infinitesimal character these polynomial identities may be expressed in a useful factored form. Our results include the characteristic identities of Bracken and Green [1] as a special case and show that these latter identities hold also in infinite dimensional representations.