What a classical r-matrix really is

Abstract

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 ${\mathcal O}$-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 $\mbox{End}\, (V) \simeq V^* \otimes V$ 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.