Dirac structures and dynamical r-matrices
Annales de l'Institut Fourier, Volume 51 (2001) no. 3, p. 835-859

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.

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.

DOI : https://doi.org/10.5802/aif.1838
Classification:  53D17,  17B62,  58H05,  70G45
Keywords: dynamical r-matrices, Dirac structures, Lie bialgebroid, Courant algebroid, lagrangian subalgebra 
@article{AIF_2001__51_3_835_0,
     author = {Liu, Zhang-Ju and Xu, Ping},
     title = {Dirac structures and dynamical $r$-matrices},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {3},
     year = {2001},
     pages = {835-859},
     doi = {10.5802/aif.1838},
     zbl = {1029.53088},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2001__51_3_835_0}
}

Liu, Zhang-Ju; Xu, Ping. Dirac structures and dynamical $r$-matrices. Annales de l'Institut Fourier, Volume 51 (2001) no. 3, pp. 835-859. doi : 10.5802/aif.1838. http://www.numdam.org/item/AIF_2001__51_3_835_0/

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