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arXiv:2206.00411 (math-ph)
[Submitted on 1 Jun 2022 (v1), last revised 30 Aug 2023 (this version, v2)]

Title:Deformations of modified $r$-matrices and cohomologies of related algebraic structures

Authors:Jun Jiang, Yunhe Sheng
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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 complement of the diagonal Lie algebra and compatible Poisson structures.
Comments: 18 pages, to appear in JNCG
Subjects: Mathematical Physics (math-ph); Rings and Algebras (math.RA)
Cite as: arXiv:2206.00411 [math-ph]
  (or arXiv:2206.00411v2 [math-ph] for this version)
  https://doi.org/10.48550/arXiv.2206.00411
Journal reference: J. Noncommut. Geom. (2024)
Related DOI: https://doi.org/10.4171/JNCG/567.

Submission history

From: Yunhe Sheng [view email]
[v1] Wed, 1 Jun 2022 11:22:47 UTC (17 KB)
[v2] Wed, 30 Aug 2023 11:33:54 UTC (18 KB)
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