What a classical r-matrix really is

Abstract

To my friend and colleague K.C. Reddy on occasion of his retirement. 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. © 1999 Taylor & Francis Group, LLC.