Uniqueness of left invariant symplectic structures on the affine Lie group
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- by Yoshio Agaoka PDF
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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 $\mathfrak {sl}(3,\mathbf {C})$.References
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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
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2753-2762
- MSC (2000): Primary 53C30, 53D05, 53D17; Secondary 17B99
- DOI: https://doi.org/10.1090/S0002-9939-01-05828-2
- MathSciNet review: 1838799