Characteristic identities for semi-simple Lie algebras

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Abstract
We present a new derivation of the polynomial identities satisfied by certain matricesAwith entriesAij(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 asp(A)= 0 wherep(x)(herein called the characteristic polynomial ofA) is a polynomial with coefficients from the centreZof the universal enveloping algebra. The minimum polynomial identitym(A)= 0 of the matrixAoverZis also obtained and it is shown thatp(x)andm(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.