Abstract
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 the complement of the diagonal Lie algebra and compatible Poisson structures.
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