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Nijenhuis operators and twisted $\mathcal{O}$-operators on Nambu-Poisson algebras

Apurba Das, Fattoum Harrathi, Sami Mabrouk

TL;DR

The paper extends the theory of Nijenhuis operators, NS-algebras, and twisted $\mathcal{O}$-operators to Nambu-Poisson algebras of order $3$, developing a second cohomology framework that combines Harrison cohomology with $3$-Lie cohomology. It studies $(1,2)$-linear deformations and proves that trivial deformations yield Nijenhuis operators, which in turn generate deformed Nambu-Poisson structures; to understand these deformations, it introduces NS-Nambu-Poisson algebras that split the NP structure. Twisted $\mathcal{O}$-operators are defined in this context and shown to encode NS-Nambu-Poisson structures via their graphs, with the induced representation carrying a natural NP structure. The work also clarifies the relationship between NS-Nambu-Poisson algebras and the subadjacent NP algebras, and identifies Reynolds operators as a specialized twisted Rota-Baxter case, linking these constructions to broader deformation and bialgebra theories. These results lay groundwork for further study of NP cohomology, NP deformations, and potential NP bialgebra formalisms, including post-NP and matched-pair constructions.

Abstract

A ternary Nambu-Poisson algebra (which we call a Nambu-Poisson algebra in the paper) is the underlying algebraic structure of Nambu-Poisson manifolds of order $3$ that appeared in the generalized Hamiltonian mechanics. First, we consider the 2nd cohomology group of a Nambu-Poisson algebra with coefficients in a given representation. Next, we discuss suitable linear deformations of a Nambu-Poisson algebra and show that any such trivial deformation yields a Nijenhuis operator on it. To understand the deformed Nambu-Poisson algebra obtained from a Nijenhuis operator, we introduce a new algebraic structure, which we name NS-Nambu-Poisson algebras. Finally, we consider $\mathcal{O}$-operators twisted by $2$-cocycles and find their close relationships with NS-Nambu-Poisson algebras.

Nijenhuis operators and twisted $\mathcal{O}$-operators on Nambu-Poisson algebras

TL;DR

The paper extends the theory of Nijenhuis operators, NS-algebras, and twisted -operators to Nambu-Poisson algebras of order , developing a second cohomology framework that combines Harrison cohomology with -Lie cohomology. It studies -linear deformations and proves that trivial deformations yield Nijenhuis operators, which in turn generate deformed Nambu-Poisson structures; to understand these deformations, it introduces NS-Nambu-Poisson algebras that split the NP structure. Twisted -operators are defined in this context and shown to encode NS-Nambu-Poisson structures via their graphs, with the induced representation carrying a natural NP structure. The work also clarifies the relationship between NS-Nambu-Poisson algebras and the subadjacent NP algebras, and identifies Reynolds operators as a specialized twisted Rota-Baxter case, linking these constructions to broader deformation and bialgebra theories. These results lay groundwork for further study of NP cohomology, NP deformations, and potential NP bialgebra formalisms, including post-NP and matched-pair constructions.

Abstract

A ternary Nambu-Poisson algebra (which we call a Nambu-Poisson algebra in the paper) is the underlying algebraic structure of Nambu-Poisson manifolds of order that appeared in the generalized Hamiltonian mechanics. First, we consider the 2nd cohomology group of a Nambu-Poisson algebra with coefficients in a given representation. Next, we discuss suitable linear deformations of a Nambu-Poisson algebra and show that any such trivial deformation yields a Nijenhuis operator on it. To understand the deformed Nambu-Poisson algebra obtained from a Nijenhuis operator, we introduce a new algebraic structure, which we name NS-Nambu-Poisson algebras. Finally, we consider -operators twisted by -cocycles and find their close relationships with NS-Nambu-Poisson algebras.
Paper Structure (11 sections, 23 theorems, 81 equations)

This paper contains 11 sections, 23 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Nambu-Poisson algebras
  3. Nijenhuis operators, NS-algebras and twisted $\mathcal{O}$-operators
  4. Overview of the main results
  5. Organization
  6. Some background on Nambu-Poisson algebras
  7. Second cohomology group, linear deformations and Nijenhuis operators
  8. The second cohomology group of a Nambu-Poisson algebra
  9. Linear deformations and Nijenhuis operators
  10. NS-Nambu-Poisson algebras
  11. Twisted $\mathcal{O}$-operators on Nambu-Poisson algebras

Key Result

Proposition 2.3

Let $(\mathcal{L}, \{ ~, ~, ~ \})$ be a $3$-Lie algebra. Suppose $V$ is a vector space with a skew-symmetric bilinear map $\rho: \mathcal{L} \times \mathcal{L} \rightarrow \mathrm{End}(V)$. Then $(V; \rho)$ is a representation of the given $3$-Lie algebra if and only if the operation for $(a, u), (b, v), (c, w) \in \mathcal{L} \oplus V$, makes $(\mathcal{L} \oplus V, \{ ~, ~, ~ \}_\ltimes)$ into

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • ...and 43 more