论文标题
相关代数结构的修改$ r $ $ $ $ $ $ $ $ $ $ $ -
Deformations of modified $r$-matrices and cohomologies of related algebraic structures
论文作者
论文摘要
修改后的$ r $ - 杂志是由Semenov-Tian-Shansky引入的修改经典Yang-baxter方程的解决方案，并且在数学物理学中扮演重要角色。在本文中，首先，我们引入了一种用于修改的$ r $ amatrices的共同体学理论。然后，我们使用已建立的共同体学理论（包括代数变形，几何变形和线性变形）研究了修改的$ r $ - amatrices的三种变形。我们给出了差异分级的谎言代数，该代数控制了修改后的$ r $ amatrices的代数变形。对于几何变形，我们证明了刚性定理和研究何时是修改的$ r $ -matrix在所有修改的$ r $ $ -matrix结构的空间中平滑的邻居。在琐碎线性变形的研究中，我们介绍了修改后的$ r $ -Matrix的Nijenhuis元素的概念。最后，用于研究对角线代数和兼容泊松结构的补体的变形。
Modified $r$-matrices are solutions of the modified classical Yang-Baxter equation, introduced by Semenov-Tian-Shansky, and play important roles in mathematical physics. In this paper, first we introduce a cohomology theory for modified $r$-matrices. Then we study three kinds of deformations of modified $r$-matrices using the established cohomology theory, including algebraic deformations, geometric deformations and linear deformations. We give the differential graded Lie algebra that governs algebraic deformations of modified $r$-matrices. For geometric deformations, we prove the rigidity theorem and study when is a neighborhood of a modified $r$-matrix smooth in the space of all modified $r$-matrix structures. In the study of trivial linear deformations, we introduce the notion of a Nijenhuis element for a modified $r$-matrix. Finally, applications are given to study deformations of complement of the diagonal Lie algebra and compatible Poisson structures.
