Solutions to the classical YBE were studied and to some extent classified by Belavin and Drinfeld. Given a 'classical r -matrix' r : V ⊗ ⊗ V → → V ⊗ ⊗ V , which may also depend on a pair of arguments ( u , v ) , the classical YBE is (suppressing parameters) [ r 12 , r 13 ] + [ r 12 , r 23 ] + [ r 13 , r 23 ] = 0. This is quadratic in the r -matrix, unlike the usual quantum YBE which is cubic in R .