Generalization of the concept of classical r-matrix to Lie algebroids

JOURNAL OF GEOMETRY AND PHYSICS

卷:165

Article

Dobrogowska, Alina¹  Jakimowicz, Grzegorz¹ 

[1] Univ Bialystok, Fac Math, Ciolkowskiego 1M, PL-15245 Bialystok, Poland

关键词: Lie algebroid;    Poisson manifold;    Lie algebra;    Classical r-matrix;    Bi-Hamiltonian structure;    Lifting of multivectors;   

DOI  :  10.1016/j.geomphys.2021.104227

来源: Elsevier

【 摘 要 】

We present some new constructions of Lie algebroids starting from vector fields on manifold M. The tangent bundle TM possess a natural structure of Lie algebroid, but we use these fields to construct a collection of interesting new algebroid structures. Next, we show that these constructions can be used in a more general situation, starting from an arbitrary Lie algebroid over M. In the final step, we show that after limiting ourselves to Lie algebras these formulas as a special case contain brackets well known in theory of classical r-matrices. We can think of our constructions as extending the concept of classical r-matrices to Lie algebroids. Several examples illustrate the importance of these constructions. (C) 2021 The Author(s). Published by Elsevier B.V.

【 授权许可】

Free