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 Zap. Nauchn. Sem. POMI, 2004, Volume 317, Pages 122–141 (Mi znsl718)

On classical $r$-matrices with parabolic carrier

V. D. Lyakhovsky

Saint-Petersburg State University

Abstract: 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.

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English version:
Journal of Mathematical Sciences (New York), 2006, 136:1, 3596–3606

Bibliographic databases:

UDC: 517.9

Citation: V. D. Lyakhovsky, “On classical $r$-matrices with parabolic carrier”, Questions of quantum field theory and statistical physics. Part 18, Zap. Nauchn. Sem. POMI, 317, POMI, St. Petersburg, 2004, 122–141; J. Math. Sci. (N. Y.), 136:1 (2006), 3596–3606

Citation in format AMSBIB
\Bibitem{Lya04} \by V.~D.~Lyakhovsky \paper On classical $r$-matrices with parabolic carrier \inbook Questions of quantum field theory and statistical physics. Part~18 \serial Zap. Nauchn. Sem. POMI \yr 2004 \vol 317 \pages 122--141 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl718} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2120834} \zmath{https://zbmath.org/?q=an:1137.17301} \transl \jour J. Math. Sci. (N. Y.) \yr 2006 \vol 136 \issue 1 \pages 3596--3606 \crossref{https://doi.org/10.1007/s10958-006-0185-8} 

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This publication is cited in the following articles:
1. J. Math. Sci. (N. Y.), 151:2 (2008), 2907–2923
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