Dirac structures and dynamical $r$-matrices

Liu, Zhang-Ju; Xu, Ping

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 $𝔤\oplus 𝔤$.

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 $𝔤\oplus 𝔤$.

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

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