THE R-MATRIX FOR NON-TWISTED AFFINE QUANTUM ALGEBRAS

This paper deals with the study of the R-matrix for non-twisted affine quantum algebras: thanks to the results of [8], [25], [26] this probl em reduces to looking for dual bases of U-q(+) and U-q(-) (dual with r espect to the Killing form). After recalling the notations (1 and 2), the general results (2 and 3) and the well known facts about the finit e case and the real root vectors (4), the first step consists in study ing the action of the coproduct on the imaginary root vectors (5 and 7 ) by applying the commutation rules found in 6. The second step is des cribing some duality properties of the PEW-bases (8), computing the Ki lling form on the imaginary root vectors (9) and finally giving comple te details on the duality properties (10). These results are gathered together in II, where an explicit exponential formula for the R-matrix is given. (C) Elsevier, Paris.