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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniqueness of left invariant symplectic structures on the affine Lie group


Author: Yoshio Agaoka
Journal: Proc. Amer. Math. Soc. 129 (2001), 2753-2762
MSC (2000): Primary 53C30, 53D05, 53D17; Secondary 17B99
Published electronically: January 23, 2001
MathSciNet review: 1838799
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Abstract:

We show the uniqueness of left invariant symplectic structures on the affine Lie group $A(n,\mathbf{R})$ under the adjoint action of $A(n,\mathbf{R})$, by giving an explicit formula of the Pfaffian of the skew symmetric matrix naturally associated with $A(n,\mathbf{R})$, 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 $SL(n,\mathbf{R})$ and $SL(n, \mathbf{C})$. This result is a generalization of Stolin's classification of constant solutions of the classical Yang-Baxter equation for $\mathfrak{sl}(2, \mathbf{C})$ and $\mathfrak{sl}(3,\mathbf{C})$.


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Additional Information

Yoshio Agaoka
Affiliation: Department of Mathematics, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan
Email: agaoka@mis.hiroshima-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05828-2
PII: S 0002-9939(01)05828-2
Keywords: Left invariant symplectic structure, affine Lie group, Pfaffian, Poisson structure, classical Yang-Baxter equation
Received by editor(s): January 12, 2000
Received by editor(s) in revised form: January 18, 2000
Published electronically: January 23, 2001
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2001 American Mathematical Society