On classical $r$-matrices with parabolic carrier

V. D. Lyakhovsky

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 18, Tome 317 (2004), pp. 122-141

Résumé

Using the graphic presentation of the dual Lie algebra $\frak{g}^{\#}(r)$ for simple algebra $\frak{g}$ it is possible to demonstrate that there always exist solutions $r_{ech}$ of the classical Yang–Baxter equation with parabolic carrier. To obtain $r_{ech}$ in the explicit form we find the dual coordinates in which the adjoint action of the carrier $\frak{g}_c$ becomes reducible. This allows to find the structure of the Jordanian $r$-matrices $r_{J}$ that are the candidates for enlarging the initial full chain $r_{fch}$ and realize the desired solution $r_{ech}$ in the factorized form $r_{ech}\approx r_{fch}+r_{J}$. We obtain the unique transformation: the canonical chain is to be substituted by a special kind of peripheric $r$-matrices: $r_{fch}\longrightarrow r_{rfch}$. To illustrate the method the case of $\frak{g}=sl(11)$ is considered in full details.