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Dirac structures and dynamical r -matrices

Zhang-Ju Liu, Ping Xu (2001)

Annales de l’institut Fourier

The purpose of this paper is to establish a connection between various objects such as dynamical r -matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical r -matrices of simple Lie algebras 𝔤 , and prove that dynamical r -matrices are in one-one correspondence with certain Lagrangian subalgebras of 𝔤 𝔤 .

On characterization of Poisson and Jacobi structures

Janusz Grabowski, Paweŀ Urbański (2003)

Open Mathematics

We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.

Poisson-Lie groupoids and the contraction procedure

Kenny De Commer (2015)

Banach Center Publications

On the level of Lie algebras, the contraction procedure is a method to create a new Lie algebra from a given Lie algebra by rescaling generators and letting the scaling parameter tend to zero. One of the most well-known examples is the contraction from 𝔰𝔲(2) to 𝔢(2), the Lie algebra of upper-triangular matrices with zero trace and purely imaginary diagonal. In this paper, we will consider an extension of this contraction by taking also into consideration the natural bialgebra structures on these...

Vector bundles on plane cubic curves and the classical Yang–Baxter equation

Igor Burban, Thilo Henrich (2015)

Journal of the European Mathematical Society

In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical r -matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic r -matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin’s classication and prove...

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