Classical r-matrices with a parabolic carrier

Using a graphic presentation of the dual Lie algebra ^# (r) for a simple algebra , it is possible to show that there always exist solutions r_ech of the classical Yang—Baxter equation with a parabolic carrier. To get a closed-form expression for r_ech, we find dual coordinates in which the adjoint action of the carrier _c is reducible. This allows us to find the structure of Jordanian r-matrices r_J which are candidates for enlarging the initial full chain r_fch and realize the desired solution r_ech in the factorized form r_ech ≈ r_fch + r_J. We obtain a unique transformation: the canonical chain has to be replaced by a special kind of peripheric r-matrices, r_fch → r_rfch. To illustrate the method, the case of =sl(11) is considered in the full detail. Bibliography: 11 titles. To my friend Petr Petrovich Kulish __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 317, 2004, pp. 122–141.