[Paper Review] Deformations of modified $r$-matrices and cohomologies of related algebraic structures

Jun Jiang, Yunhe Sheng|arXiv (Cornell University)|Jun 1, 2022

Advanced Topics in Algebra28 references4 citations

TL;DR

This paper introduces a cohomology theory for modified $r$-matrices and develops three deformation theories—algebraic, geometric, and linear—using this framework. It establishes a differential graded Lie algebra governing algebraic deformations, proves rigidity and smoothness conditions via cohomology vanishing, and introduces Nijenhuis elements for trivial linear deformations. Key results include a cohomological characterization of deformations and applications to matched pairs and compatible Poisson structures.

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.

Motivation & Objective

• To develop a cohomology theory for modified $r$-matrices using the Chevalley-Eilenberg complex of the induced Lie algebra $\mathfrak{g}_R$.
• To study algebraic, geometric, and linear deformations of modified $r$-matrices within a unified cohomological framework.
• To establish a differential graded Lie algebra that governs algebraic deformations, fulfilling the Deligne-Drinfeld-Kontsevich slogan.
• To prove rigidity and smoothness theorems for geometric deformations using vanishing of $H^2(R)$ and $H^3(R)$.
• To introduce the notion of Nijenhuis elements for trivial linear deformations and relate them to Nijenhuis operators on $\mathfrak{g}_R$.

Proposed method

• Define the cohomology of a modified $r$-matrix $R$ as the Chevalley-Eilenberg cohomology of the Lie algebra $\mathfrak{g}_R$ with coefficients in $\mathfrak{g}$, where $[x,y]_R = [R(x),y]_\mathfrak{g} + [x,R(y)]_\mathfrak{g}$.
• Construct a differential graded Lie algebra (DGLA) that controls algebraic deformations of $R$, using the cohomology complex and graded Lie bracket.
• Analyze geometric deformations $R_t$ via the orbit method, identifying the tangent space to the orbit $\mathrm{Orb}_R$ as the space of 2-coboundaries $B^2(R)$.
• Use the Kuranishi map to characterize necessary and sufficient conditions for a 2-cocycle to yield a geometric deformation.
• Define linear deformations $R_t = R + t\hat{R}$ and identify trivial deformations via Nijenhuis elements $x \in \mathfrak{g}$ such that $\mathrm{ad}_x$ is a Nijenhuis operator on $\mathfrak{g}_R$.
• Apply the theory to two settings: deformations of complements in $\mathfrak{g} \oplus \mathfrak{g}$ and compatible Poisson structures on $\mathfrak{g}^*$.

Experimental results

Research questions

• RQ1 How can a cohomology theory be defined for modified $r$-matrices, and how does it relate to the cohomology of Rota-Baxter operators?
• RQ2 What differential graded Lie algebra governs algebraic deformations of modified $r$-matrices?
• RQ3 Under what cohomological conditions is a modified $r$-matrix rigid or locally smooth in the space of all such structures?
• RQ4 What is the role of Nijenhuis elements in classifying trivial linear deformations of modified $r$-matrices?
• RQ5 How do deformations of modified $r$-matrices induce compatible Poisson structures on the dual space $\mathfrak{g}^*$?

Key findings

• The cohomology of a modified $r$-matrix $R = \mathrm{Id} + 2B$ is isomorphic to the cohomology of the corresponding Rota-Baxter operator $B$ of weight 1.
• Algebraic deformations of $R$ are governed by a differential graded Lie algebra, confirming the general deformation-theoretic slogan.
• If $H^2(R) = 0$, then $R$ is rigid under geometric deformations; if $H^3(R) = 0$, then the space of modified $r$-matrices is a smooth manifold near $R$.
• A 2-cocycle $\hat{R}$ gives rise to a geometric deformation if and only if it satisfies the Kuranishi condition via the Kuranishi map.
• A linear deformation $R_t = R + t\hat{R}$ is trivial if and only if $\hat{R}$ corresponds to a Nijenhuis element $x \in \mathfrak{g}$, with $\mathrm{ad}_x$ a Nijenhuis operator on $\mathfrak{g}_R$.
• For any $t_1, t_2 \in \mathbb{R}$, the Poisson structures $\{\cdot,\cdot\}_{R_{t_1}}$ and $\{\cdot,\cdot\}_{R_{t_2}}$ on $\mathfrak{g}^*$ are compatible, as $R_{t_1} + R_{t_2}$ is also a modified $r$-matrix.