FROM RATIONAL TO TRIGONOMETRIC R-MATRICES

The skew-symmetric trigonometric R-matrices in the classification of Belavin and Drinfel'd are shown to be particular elements of a linear set of R-matrices arising from the deformation of the canonical non-skew-symmetric q-pole R-matrix belonging to the Lie algebra gxgxC(lambda, mu). This deformation is consistent only when the Lie algebra automorphism defining the q-pole R-matrix is a Coxeter automorphism. This linear deformation is the only one allowed when the algebraic structure of R as an element of gxg is required to be preserved, at least when g = sl (2, C).