Classical Yang-Baxter equation and left invariant affine geometry on lie groups

By A. Diatta, A. Medina. 2002

61 citations recorded by the source corpus.

Abstract.Let G be a Lie group, T*G=Lie(G)*⋊G its cotangent bundle considered as a Lie group, where G acts on Lie(G)* via the coadjoint action. Each solution r of the Classical Yang Baxter Equation on G, corresponds to a connected Lie subgroup H of T*G such that Lie(H) is a Lagrangian graph in Lie(G)⊕Lie(G)* and H carries a left invariant affine structure. If r is invertible, the Poisson Lie tensor π given by r on G is polynomial of degree at most 2 and every double Lie group of (G,π) is endowed with an affine and a complex structures ∇ and J, both left invariant and given by r, such that ∇J=0.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/468a04a06adcc3e77fbd1e9e9c97acf0d41a57ed

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In Applied Mathematics

1. Li Guo
2. Chengming Bai
3. S. Salamon
4. Adrián Andrada
5. Anna Fino

In Geometry and Topology

1. Li Guo
2. Chengming Bai
3. Xianglong Ni
4. Adrián Andrada
5. S. Salamon
6. Kurusch Ebrahimi-Fard
7. William F. Keigher
8. Anna Fino

In Statistical and Nonlinear Physics

1. Li Guo
2. Chengming Bai
3. S. Salamon

Citations in the corpus

• Complex product structures on Lie algebras (2003, background): “3 of [11], we obtain that the product defined above determines an LSA structure on g if and only if Φ ≡ 0 and Ψ ≡ 0.”
• Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras (2009, background): “(ii) The complex structure J−1,0 has already been found in [ 16 ].”