On Classical $r$-Matrix for the Kowalevski Gyrostat on $\mathrm{so}(4)$
Symmetry, integrability and geometry: methods and applications, Tome 2 (2006) Cet article a éte moissonné depuis la source Math-Net.Ru

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We present the trigonometric Lax matrix and classical $r$-matrix for the Kowalevski gyrostat on $\mathrm{so}(4)$ algebra by using the auxiliary matrix algebras $\mathrm{so}(3,2)$ or $\mathrm{sp}(4)$.
Keywords: Kowalevski top; Lax matrices; classical $r$-matrix.
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Igor V. Komarov; Andrey V. Tsiganov. On Classical $r$-Matrix for the Kowalevski Gyrostat on $\mathrm{so}(4)$. Symmetry, integrability and geometry: methods and applications, Tome 2 (2006). http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a11/

[1] Babelon O., Viallet C. M., “Hamiltonian structures and Lax equations”, Phys. Lett. B, 237 (1990), 411–416 | DOI | MR

[2] Belavin A. A., Drinfeld V. G., “Solutions of the classical Yang–Baxter equation for simple Lie algebras”, Funct. Anal. Appl., 16:3 (1982), 159–180 ; Etingof P., Varchenko A., “Geometry and classification of solutions of the classical dynamical Yang–Baxter equation”, Comm. Math. Phys., 192 (1998), 77–120 ; arXiv:q-alg/9703040 | DOI | MR | DOI | MR | Zbl

[3] Bobenko A. I., Kuznetsov V. B., “Lax representation for the Goryachev–Chaplygin top and new formulae for its solutions”, J. Phys. A: Math. Gen., 21 (1988), 1999–2006 ; Sklyanin E. K., “Dynamical $r$-matrices for the elliptic Calogero–Moser model”, Algebra i Analiz, 6:2 (1994), 227–237 | DOI | MR | Zbl | MR | Zbl

[4] St. Petersburg Math. J., 6:2 (1995), 397–406 | DOI | MR | MR | Zbl | Zbl

[5] Kowalevski S., “Sur le probléme de la rotation d'un corps solide autour d'un point fixe”, Acta Math., 12 (1889), 177–232 | DOI | MR

[6] Komarov I. V., Sokolov V. V., Tsiganov A. V., “Poisson maps and integrable deformations of Kowalevski top”, J. Phys. A: Math. Gen., 36 (2003), 8035–8048 ; arXiv:nlin.SI/0304033 | DOI | MR | Zbl

[7] Marshall I. D., “The Kowalevski top: its $r$-matrix interpretation and bi-Hamiltonian formulation”, Comm. Math. Phys., 191 (1998), 723–734 | DOI | MR | Zbl

[8] Reyman A. G., Semenov-Tian-Shansky M. A., “Group-theoretical methods in the theory of finite-dimensional integrable systems”, Dynamical Systems, VII, Encyclopaedia of Mathematical Sciences, 16, eds. V. I. Arnold and S. P. Novikov, Springer, Berlin, 1994, 116–225 | MR

[9] Reyman A. G., Semenov-Tian-Shansky M. A., “Lax representation with a spectral parameter for the Kowalewski top and its generalizations”, Lett. Math. Phys., 14 (1987), 55–61 | DOI | MR | Zbl

[10] Theor. Math. Phys., 141:1 (2004), 1348–1360 | DOI | MR | Zbl