Casimir invariants and vector operators in simple and classical Lie algebras

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Abstract
A method of computing eigenvalues of certain types of Casimir invariants has been developed for simple and classical Lie algebras. Especially these eigenvalues for algebras An, Bn, Cn, Dn, and G2 have been computed in closed terms. We also enumerate numbers and functional forms of all linearly independent vector operators in terms of generators in any irreducible representation of these algebras. Some polynomial identities among infinitesimal generators of these algebras are derived by means of the same technique.