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

Jun Jiang, Yunhe Sheng

TL;DR

The paper develops a cohomology theory for modified $r$-matrices via the Chevalley–Eilenberg complex of the associated Lie algebra $(\mathfrak g)_R$ and connects it to Rota–Baxter theory through $R={\rm Id}+2B$. It then constructs a differential graded Lie algebra governing algebraic deformations, and uses Crainic–Schatz–Struchiner techniques to analyze geometric deformations, establishing rigidity and local manifold structure under appropriate cohomological vanishing. Linear deformations are linked to 2-cocycles and 2nd cohomology classes, with Nijenhuis elements producing trivial deformations and acting as generators of Nijenhuis operators on the deformed Lie algebra. Applications include deformations of the diagonal complement in doubles and the compatibility of Poisson structures on the dual, illustrating the utility of the cohomology and DG-Lie framework for concrete deformation problems in Lie theory and Poisson geometry.

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.

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

TL;DR

The paper develops a cohomology theory for modified -matrices via the Chevalley–Eilenberg complex of the associated Lie algebra and connects it to Rota–Baxter theory through . It then constructs a differential graded Lie algebra governing algebraic deformations, and uses Crainic–Schatz–Struchiner techniques to analyze geometric deformations, establishing rigidity and local manifold structure under appropriate cohomological vanishing. Linear deformations are linked to 2-cocycles and 2nd cohomology classes, with Nijenhuis elements producing trivial deformations and acting as generators of Nijenhuis operators on the deformed Lie algebra. Applications include deformations of the diagonal complement in doubles and the compatibility of Poisson structures on the dual, illustrating the utility of the cohomology and DG-Lie framework for concrete deformation problems in Lie theory and Poisson geometry.

Abstract

Modified -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 -matrices. Then we study three kinds of deformations of modified -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 -matrices. For geometric deformations, we prove the rigidity theorem and study when is a neighborhood of a modified -matrix smooth in the space of all modified -matrix structures. In the study of trivial linear deformations, we introduce the notion of a Nijenhuis element for a modified -matrix. Finally, applications are given to study deformations of complement of the diagonal Lie algebra and compatible Poisson structures.
Paper Structure (8 sections, 23 theorems, 75 equations)

This paper contains 8 sections, 23 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Cohomologies of modified $r$-matrices
  3. Algebraic deformations of modified $r$-matrices
  4. Geometric deformations of modified $r$-matrices
  5. Linear deformations of modified $r$-matrices
  6. Applications
  7. Deformations of complements
  8. Compatible Poisson structures

Key Result

Proposition 2.5

Let $R$ be a modified $r$-matrix on a Lie algebra $(\mathfrak g, [\cdot, \cdot]_\mathfrak g)$. Define a linear map $\rho:\mathfrak g\,\rightarrow\,\mathfrak {gl}(\mathfrak g)$ by Then $\rho$ is a representation of the Lie algebra $(\mathfrak g, [\cdot,\cdot]_R)$ on the vector space $\mathfrak g$.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • ...and 48 more