## Uniqueness of left invariant symplectic structures on the affine Lie group.(English)Zbl 1021.53052

Summary: We show the uniqueness of left invariant symplectic structures on the affine Lie group $$A(n,{\mathbb{R}})$$ under the adjoint action of $$A(n,{\mathbb{R}})$$, by giving an explicit formula of the Pfaffian of the skew symmetric matrix naturally associated with $$A(n,{\mathbb{R}})$$, and also by giving an unexpected identity on it which relates two left invariant symplectic structures. As an application of this result, we classify maximum rank left invariant Poisson structures on the simple Lie groups $$SL(n,{\mathbb{R}})$$ and $$SL(n, {\mathbb{C}})$$. This result is a generalization of Stolin’s classification of constant solutions of the classical Yang-Baxter equation for $$\mathfrak{sl}(2, {\mathbb{C}})$$ and $$\mathfrak{sl}(3,{\mathbb{C}})$$.

### MSC:

 53D05 Symplectic manifolds (general theory) 53D17 Poisson manifolds; Poisson groupoids and algebroids 53C30 Differential geometry of homogeneous manifolds 17B63 Poisson algebras
Full Text:

### References:

 [1] Y. Agaoka, Left invariant Poisson structures on classical non-compact simple Lie groups, Israel J. Math. 116 (2000), 189-222. · Zbl 0951.22010 [2] A. A. Belavin and V. G. Drinfel$$^{\prime}$$d, Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funktsional. Anal. i Prilozhen. 16 (1982), no. 3, 1 – 29, 96 (Russian). [3] Martin Bordemann, Alberto Medina, and Ali Ouadfel, Le groupe affine comme variété symplectique, Tohoku Math. J. (2) 45 (1993), no. 3, 423 – 436 (French, with English summary). · Zbl 0784.53031 [4] Michel Cahen, Simone Gutt, and John Rawnsley, Some remarks on the classification of Poisson Lie groups, Symplectic geometry and quantization (Sanda and Yokohama, 1993) Contemp. Math., vol. 179, Amer. Math. Soc., Providence, RI, 1994, pp. 1 – 16. · Zbl 0820.58018 [5] Jean-Michel Dardié and Alberto Medina, Double extension symplectique d’un groupe de Lie symplectique, Adv. Math. 117 (1996), no. 2, 208 – 227 (French, with English summary). · Zbl 0843.58045 [6] E. B. Dynkin, The maximal subgroups of the classical groups, Amer. Math. Soc. Translation Ser.2, Vol. 6 (1957), 245-378. · Zbl 0077.03403 [7] Zhang Ju Liu and Ping Xu, On quadratic Poisson structures, Lett. Math. Phys. 26 (1992), no. 1, 33 – 42. · Zbl 0773.58007 [8] Alberto Medina and Philippe Revoy, Groupes de Lie à structure symplectique invariante, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 20, Springer, New York, 1991, pp. 247 – 266 (French). · Zbl 0754.53027 [9] Kentaro Mikami, Symplectic and Poisson structures on some loop groups, Symplectic geometry and quantization (Sanda and Yokohama, 1993) Contemp. Math., vol. 179, Amer. Math. Soc., Providence, RI, 1994, pp. 173 – 192. · Zbl 0820.58021 [10] Alfons I. Ooms, On Lie algebras having a primitive universal enveloping algebra, J. Algebra 32 (1974), no. 3, 488 – 500. · Zbl 0355.17014 [11] Alfons I. Ooms, On Frobenius Lie algebras, Comm. Algebra 8 (1980), no. 1, 13 – 52. · Zbl 0421.17004 [12] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1 – 155. · Zbl 0321.14030 [13] A. Stolin, Constant solutions of Yang-Baxter equation for \?\?(2) and \?\?(3), Math. Scand. 69 (1991), no. 1, 81 – 88. · Zbl 0727.17006 [14] Izu Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. · Zbl 0810.53019
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.