Key points are not available for this paper at this time.
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.
Jiang et al. (Tue,) studied this question.