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://www.numdam.org/articles/10.5802/aif.1838/

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