## Dirac structures and dynamical $$r$$-matrices.(English)Zbl 1029.53088

Summary: 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 $${\mathfrak g}\oplus{\mathfrak g}$$.

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids 17B62 Lie bialgebras; Lie coalgebras 58H05 Pseudogroups and differentiable groupoids 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
Full Text:

### References:

 [1] Invariant varieties through singularities of holomorphic vector fields, Annals of Math. (2), 115, (1982) · Zbl 0503.32007 [2] Universal solutions of quantum dynamical Yang-Baxter equation, Lett. Math. Phys., 44, 201-214, (1998) · Zbl 0973.81047 [3] Classical dynamical r-matrices for Calogero-Moser systems and their generalizations, q-alg/9706024 [4] Equation de Yang-Baxter dynamique classique et algebroïdes de Lie, C. R. Acad. Sci. Paris, Série I, 327, 541-546, (1998) · Zbl 0973.58007 [5] Triangle equations and simple Lie algebras, Math. Phys. Review, 4, 93-165, (1984) · Zbl 0553.58040 [6] The r-matrix structure of the Euler-Calogero-Moser model, Phys. Lett. A, 186, 114-118, (1994) · Zbl 0941.37514 [7] Exact Yangian symmetry in the classical Euler-Calogero-Moser model, Phys. Lett. A, 188, 263-271, (1994) · Zbl 0941.37512 [8] Dirac manifolds, Trans. A.M.S., 319, 631-661, (1990) · Zbl 0850.70212 [9] Quasi-Hopf algebras, Leningrad Math. J., 2, 829-860, (1991) · Zbl 0728.16021 [10] On Poisson homogeneous spaces of Poisson-Lie groups, Theor. Math. Phys., 95, 524-525, (1993) · Zbl 0852.22018 [11] Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys., 192, 77-120, (1998) · Zbl 0915.17018 [12] Conformal field theory and integrable systems associated to elliptic curves, Proc. Int. Congr. Math. Zürich, 1247-1255, (1994), Birkhäuser, Basel · Zbl 0852.17014 [13] Quasi-Hopf deformation of quantum groups, Lett. Math. Phys., 40, 117-134, (1997) · Zbl 0882.17006 [14] Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups, 4, 303-327, (1999) · Zbl 0977.17012 [15] Poisson homogeneous spaces of Poisson-Lie groups, (1997) [16] Exact gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math., 41, 153-165, (1995) · Zbl 0837.17014 [17] Some remarks on Dirac structures and Poisson reductions, Banach Center Publ., 51, 165-173, (2000) · Zbl 0966.58013 [18] Manin triples for Lie bialgebroids, J. Diff. Geom., 45, 547-574, (1997) · Zbl 0885.58030 [19] Dirac structures and Poisson homogeneous spaces, Comm. Math. Phys., 192, 121-144, (1998) · Zbl 0921.58074 [20] Exact Lie bialgebroids and Poisson groupoids, Geom. Funct. Anal., 6, 138-145, (1996) · Zbl 0869.17016 [21] Classical dynamical $$r$$-matrices and homogeneous Poisson structures on $$G/H$$ and $$K/T,$$ Comm. Math. Phys., 212, 337-370, (2000) · Zbl 1008.53064 [22] Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Diff. Geom., 31, 501-526, (1990) · Zbl 0673.58018 [23] Lie bialgebroids and Poisson groupoids, Duke Math. J., 18, 415-452, (1994) · Zbl 0844.22005 [24] Integration of Lie bialgebroids, Topology, 39, 445-467, (2000) · Zbl 0961.58009 [25] Dressing transformations and Poisson Lie group actions, 21, 1237-1260, (1985), Publ. RIMS, Kyoto University · Zbl 0674.58038 [26] On classification of dynamical $$r$$-matrices, Math. Res. Lett., 5, 13-30, (1998) · Zbl 0957.17020 [27] Poisson geometry, Diff. Geom. Appl., 9, 213-238, (1998) · Zbl 0930.37032 [28] Quantum groupoids associated to universal dynamical R-matrices, C. R. Acad. Sci. Paris, Série I, 328, 327-332, (1999) · Zbl 0939.17013 [29] Quantum groupoids, Comm. Math. Phys., 216, 539-581, (2001) · Zbl 0986.17003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.