What a Classical r-Matrix Really Is

The notion of classical r-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, — where the standard definitions are shown to be deficient, — is proposed, the notion of an O-operator. This notion has all the natural properties one would expect form it, but lacks those which are artifacts of finite-dimensional isomorpisms such as not true in differential generality relation End (V) ≃ V* ⊗ V for a vector space V. Examples considered include a quadratic Poisson bracket on the dual space to a Lie algebra; generalized symplectic-quadratic models of such brackets (aka Clebsch representations); and Drinfel'd's 2-cocycle interpretation of nondegenate classical r-matrices.