Dirac structures and dynamical $r$-matrices

[Structures de Dirac et $r$-matrices dynamiques]

Liu, Zhang-Ju ; Xu, Ping

Annales de l'Institut Fourier, Tome 51 (2001), p. 835-859 / Harvested from Numdam

Résumé

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 ${\mathfrak {g}}$, et prouvons que ces $r$-matrices dynamiques sont en bijection avec certaines sous-algèbres lagrangiennes de ${\mathfrak {g}} \oplus {\mathfrak {g}}$.

Abstract

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 ${\mathfrak {g}}$, and prove that dynamical $r$-matrices are in one-one correspondence with certain Lagrangian subalgebras of ${\mathfrak {g}}\oplus {\mathfrak {g}}$.

Détail

Publié le : 2001-01-01

Zbl 1029.53088 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1838
Classification:  53D17,  17B62,  58H05,  70G45
Mots clés: $r$-matrice dynamique, structure de Dirac, bialgébroïde de Lie, algébroïde de Courant, sous-algèbre lagrangienne

Bibliographie

[1] C. Camacho; P. Sad Invariant varieties through singularities of holomorphic vector fields, Annals of Math. (2), Tome 115 (1982) | Zbl 0503.32007

[1] D. Arnaudon; E. Buffenoir; E. Ragoucy; And Ph. Roche Universal solutions of quantum dynamical Yang-Baxter equation, Lett. Math. Phys., Tome 44 (1998), pp. 201-214 | Article | Zbl 0973.81047

[2] J. Avan Classical dynamical r-matrices for Calogero-Moser systems and their generalizations, Tome 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, Tome 327 (1998), pp. 541-546 | Zbl 0973.58007

[4] A. Belavin; And V. Drinfeld Triangle equations and simple Lie algebras, Math. Phys. Review, Tome 4 (1984), pp. 93-165 | Zbl 0553.58040

[5] E. Billey; J. Avan; And O. Babelon The r-matrix structure of the Euler-Calogero-Moser model, Phys. Lett. A, Tome 186 (1994), pp. 114-118 | Article | Zbl 0941.37514

[6] E. Billey; J. Avan; And O. Babelon Exact Yangian symmetry in the classical Euler-Calogero-Moser model, Phys. Lett. A, Tome 188 (1994), pp. 263-271 | Article | Zbl 0941.37512

[7] T.-J. Courant Dirac manifolds, Trans. A.M.S., Tome 319 (1990), pp. 631-661 | Article | Zbl 0850.70212

[8] V.-G. Drinfel'D Quasi-Hopf algebras, Leningrad Math. J., Tome 2 (1991), pp. 829-860 | Zbl 0728.16021

[9] V.-G. Drinfel'D On Poisson homogeneous spaces of Poisson-Lie groups, Theor. Math. Phys., Tome 95 (1993), p. 524-525 | Article | Zbl 0852.22018

[10] P. Etingof; And A. Varchenko Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys., Tome 192 (1998), pp. 77-120 | Article | Zbl 0915.17018

[11] G. Felder Conformal field theory and integrable systems associated to elliptic curves, Proc. Int. Congr. Math. Zürich, Birkhäuser, Basel (1994), pp. 1247-1255 | Zbl 0852.17014

[12] C. Fronsdal Quasi-Hopf deformation of quantum groups, Lett. Math. Phys., Tome 40 (1997), pp. 117-134 | Article | Zbl 0882.17006

[13] M. Jimbo; H. Konno; Odake; And J. Shiraishi Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups, Tome 4 (1999), pp. 303-327 | Article | Zbl 0977.17012

[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., Tome 41 (1995), pp. 153-165 | Article | Zbl 0837.17014

[16] Z.-J. Liu Some remarks on Dirac structures and Poisson reductions, Banach Center Publ., Tome 51 (2000), pp. 165-173 | Zbl 0966.58013

[17] Z.-J. Liu; A. Weinstein; P. Xu Manin triples for Lie bialgebroids, J. Diff. Geom., Tome 45 (1997), pp. 547-574 | Zbl 0885.58030

[18] Z.-J. Liu; A. Weinstein; P. Xu Dirac structures and Poisson homogeneous spaces, Comm. Math. Phys., Tome 192 (1998), pp. 121-144 | Article | Zbl 0921.58074

[19] Z.-J. Liu; P. Xu Exact Lie bialgebroids and Poisson groupoids, Geom. Funct. Anal., Tome 6 (1996), pp. 138-145 | Article | Zbl 0869.17016

[20] J.-H. Lu Classical dynamical $r$-matrices and homogeneous Poisson structures on $G/H$ and $K/T$, Comm. Math. Phys., Tome 212 (2000), pp. 337-370 | Article | Zbl 1008.53064

[21] J.-H. Lu; A. Weinstein Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Diff. Geom., Tome 31 (1990), pp. 501-526 | Zbl 0673.58018

[22] K. Mackenzie; And P. Xu Lie bialgebroids and Poisson groupoids, Duke Math. J., Tome 18 (1994), pp. 415-452 | Article | Zbl 0844.22005

[23] K. Mackenzie; And P. Xu Integration of Lie bialgebroids, Topology, Tome 39 (2000), pp. 445-467 | Article | Zbl 0961.58009

[24] M.-A. Semenov-Tian-Shansky Dressing transformations and Poisson Lie group actions, Publ. RIMS, Kyoto University, Tome 21 (1985), pp. 1237-1260 | Zbl 0674.58038

[25] O. Schiffmann On classification of dynamical $r$-matrices, Math. Res. Lett., Tome 5 (1998), pp. 13-30 | Zbl 0957.17020

[26] A. Weinstein Poisson geometry, Diff. Geom. Appl., Tome 9 (1998), pp. 213-238 | Article | Zbl 0930.37032

[27] P. Xu Quantum groupoids associated to universal dynamical R-matrices, C. R. Acad. Sci. Paris, Série I, Tome 328 (1999), pp. 327-332 | Zbl 0939.17013

[28] P. Xu Quantum groupoids, Comm. Math. Phys., Tome 216 (2001), pp. 539-581 | Article | Zbl 0986.17003