Uniqueness of left invariant symplectic structures on the affine Lie group

Abstract

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