Dirac structures and dynamical r-matrices
[Structures de Dirac et r-matrices dynamiques]
Liu, Zhang-Ju1 ; Xu, Ping2
1 Peking University, Department of Mathematics, Beijing 100871 (Rép. Pop. Chine)
2 Pennsylvania State University, Department of Mathematics, University Park PA 16802 (USA)
Annales de l'Institut Fourier, Tome 51 (2001) no. 3, pp. 835-859.

Le but de cet article est d’établir un lien entre différents sujets tels que les r- matrices dynamiques, les bialgèbroïdes de Lie et les sous-algèbres lagrangiennes. Notre méthode se base sur la théorie des structures de Dirac et algébroïdes de Courant. En particulier, nous donnons une nouvelle méthode pour classifier les r-matrices dynamiques des algèbres de Lie simples 𝔤, et prouvons que ces r-matrices dynamiques sont en bijection avec certaines sous-algèbres lagrangiennes de 𝔤⊕𝔤.

The purpose of this paper is to establish a connection between various objects such as dynamical r-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical r-matrices of simple Lie algebras 𝔤, and prove that dynamical r-matrices are in one-one correspondence with certain Lagrangian subalgebras of 𝔤⊕𝔤.

DOI : 10.5802/aif.1838
Classification : 53D17, 17B62, 58H05, 70G45
Keywords: dynamical $r$-matrices, Dirac structures, Lie bialgebroid, Courant algebroid, lagrangian subalgebra
Mot clés : $r$-matrice dynamique, structure de Dirac, bialgébroïde de Lie, algébroïde de Courant, sous-algèbre lagrangienne
Liu, Zhang-Ju 1 ; Xu, Ping 2

1 Peking University, Department of Mathematics, Beijing 100871 (Rép. Pop. Chine)
2 Pennsylvania State University, Department of Mathematics, University Park PA 16802 (USA) 
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Liu, Zhang-Ju; Xu, Ping. Dirac structures and dynamical $r$-matrices. Annales de l'Institut Fourier, Tome 51 (2001) no. 3, pp. 835-859. doi : 10.5802/aif.1838. http://archive.numdam.org/articles/10.5802/aif.1838/

[1] C. Camacho; P. Sad Invariant varieties through singularities of holomorphic vector fields, Annals of Math. (2), Volume 115 (1982) | Zbl

[1] D. Arnaudon; E. Buffenoir; E. Ragoucy; and Ph. Roche Universal solutions of quantum dynamical Yang-Baxter equation, Lett. Math. Phys., Volume 44 (1998), pp. 201-214 | DOI | Zbl

[2] J. Avan Classical dynamical r-matrices for Calogero-Moser systems and their generalizations, Volume q-alg/9706024

[3] M. Bangoura; and Y. Kosmann-Schwarzbach Equation de Yang-Baxter dynamique classique et algebroïdes de Lie, C. R. Acad. Sci. Paris, Série I, Volume 327 (1998), pp. 541-546 | Zbl

[4] A. Belavin; and V. Drinfeld Triangle equations and simple Lie algebras, Math. Phys. Review, Volume 4 (1984), pp. 93-165 | Zbl

[5] E. Billey; J. Avan; and O. Babelon The r-matrix structure of the Euler-Calogero-Moser model, Phys. Lett. A, Volume 186 (1994), pp. 114-118 | DOI | Zbl

[6] E. Billey; J. Avan; and O. Babelon Exact Yangian symmetry in the classical Euler-Calogero-Moser model, Phys. Lett. A, Volume 188 (1994), pp. 263-271 | DOI | Zbl

[7] T.-J. Courant Dirac manifolds, Trans. A.M.S., Volume 319 (1990), pp. 631-661 | DOI | Zbl

[8] V.-G. Drinfel'd Quasi-Hopf algebras, Leningrad Math. J., Volume 2 (1991), pp. 829-860 | Zbl

[9] V.-G. Drinfel'd On Poisson homogeneous spaces of Poisson-Lie groups, Theor. Math. Phys., Volume 95 (1993), pp. 524-525 | DOI | Zbl

[10] P. Etingof; and A. Varchenko Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys., Volume 192 (1998), pp. 77-120 | DOI | Zbl

[11] G. Felder Conformal field theory and integrable systems associated to elliptic curves, Proc. Int. Congr. Math. Zürich (1994), pp. 1247-1255 | Zbl

[12] C. Fronsdal Quasi-Hopf deformation of quantum groups, Lett. Math. Phys., Volume 40 (1997), pp. 117-134 | DOI | Zbl

[13] M. Jimbo; H. Konno; Odake; and J. Shiraishi Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups, Volume 4 (1999), pp. 303-327 | DOI | Zbl

[14] E. Karolinsky Poisson homogeneous spaces of Poisson-Lie groups (1997) (Ph. D. thesis, The institute of low temperature, Kharkov)

[15] Y. Kosmann-Schwarzbach Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math., Volume 41 (1995), pp. 153-165 | DOI | Zbl

[16] Z.-J. Liu Some remarks on Dirac structures and Poisson reductions, Banach Center Publ., Volume 51 (2000), pp. 165-173 | Zbl

[17] Z.-J. Liu; A. Weinstein; P. Xu Manin triples for Lie bialgebroids, J. Diff. Geom., Volume 45 (1997), pp. 547-574 | Zbl

[18] Z.-J. Liu; A. Weinstein; P. Xu Dirac structures and Poisson homogeneous spaces, Comm. Math. Phys., Volume 192 (1998), pp. 121-144 | DOI | Zbl

[19] Z.-J. Liu; P. Xu Exact Lie bialgebroids and Poisson groupoids, Geom. Funct. Anal., Volume 6 (1996), pp. 138-145 | DOI | Zbl

[20] J.-H. Lu Classical dynamical r-matrices and homogeneous Poisson structures on G/H and K/T, Comm. Math. Phys., Volume 212 (2000), pp. 337-370 | DOI | Zbl

[21] J.-H. Lu; A. Weinstein Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Diff. Geom., Volume 31 (1990), pp. 501-526 | Zbl

[22] K. Mackenzie; and P. Xu Lie bialgebroids and Poisson groupoids, Duke Math. J., Volume 18 (1994), pp. 415-452 | DOI | Zbl

[23] K. Mackenzie; and P. Xu Integration of Lie bialgebroids, Topology, Volume 39 (2000), pp. 445-467 | DOI | Zbl

[24] M.-A. Semenov-Tian-Shansky Dressing transformations and Poisson Lie group actions, Volume 21 (1985), pp. 1237-1260 | Zbl

[25] O. Schiffmann On classification of dynamical r-matrices, Math. Res. Lett., Volume 5 (1998), pp. 13-30 | Zbl

[26] A. Weinstein Poisson geometry, Diff. Geom. Appl., Volume 9 (1998), pp. 213-238 | DOI | Zbl

[27] P. Xu Quantum groupoids associated to universal dynamical R-matrices, C. R. Acad. Sci. Paris, Série I, Volume 328 (1999), pp. 327-332 | Zbl

[28] P. Xu Quantum groupoids, Comm. Math. Phys., Volume 216 (2001), pp. 539-581 | DOI | Zbl

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