Casimir invariants and characteristic identities for $gl(\infty )$
  A full set of (higher order) Casimir invariants for the Lie algebra $gl(\infty )$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight (unitarizable) irreducible representations with only a finite number of non-zero weight components. Moreover the eigenvalues of these Casimir invariants are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from $gl(\infty )$ are also determined and generalize those previously obtained for $gl(n)$ by Bracken and Green.$^{1,2}$ 
M.D. Gould, N.I. Stoilova