[Paper Review] Associative and Lie deformations of Poisson algebras

Elisabeth Remm|arXiv (Cornell University)|May 13, 2011

Advanced Topics in Algebra14 references3 citations

TL;DR

This paper reinterprets deformations of Poisson algebras by treating them as deformations of a single nonassociative multiplication satisfying the Markl-Remm identity, unifying associative and Lie deformations under a common cohomological framework. The key contribution is the identification of Poisson cohomology as the governing theory for general deformations, with Poisson-Lichnerowicz and Poisson-Hochschild cohomologies as special cases, and a precise characterization of when a cochain is closed under the Poisson coboundary operator via skew-symmetric and symmetric derivations.

ABSTRACT

Considering a Poisson algebra as a non associative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this non associative algebra. This gives a natural interpretation of deformations which preserves the underlying associative structure and we study deformations which preserve the underlying Lie algebra.

Motivation & Objective

• To provide a unified framework for deformations of Poisson algebras by treating them as deformations of a single nonassociative multiplication satisfying the Markl-Remm identity.
• To recover classical Lie deformations (preserving the associative product) and associative deformations (preserving the Lie bracket) as special cases within a single deformation theory.
• To define and study Poisson-Hochschild cohomology as the parameter space for associative deformations.
• To clarify the relationship between Poisson cohomology, Poisson-Lichnerowicz cohomology, and Poisson-Hochschild cohomology through explicit coboundary operators and decomposition formulas.

Proposed method

• Represent a Poisson algebra as a nonassociative algebra via the Markl-Remm identity, encoding both the associative product and Lie bracket in a single multiplication.
• Define the Poisson cohomology complex using a coboundary operator δⁿ_P acting on multilinear cochains, with explicit formulas involving symmetric and skew-symmetric parts.
• Decompose a cochain φ into its symmetric part φ_s and skew-symmetric part φ_a using projection operators Φ_{W_n} and Φ_{V_n} respectively.
• Introduce operators L_{1,n} and L_{H,n} to characterize when a skew-symmetric cochain is a derivation for the associative product.
• Define the operator ∇ⁿ to characterize symmetric cochains as Lie n-derivations, and relate it to the Poisson coboundary via ∇ⁿφ_s = 0 if and only if δ_Hⁿφ_s = 0.
• Use the identity δ_Pⁿφ = 0 if and only if δ_Cⁿφ_a = 0 and L_{1,n}φ_a = 0 to show that closed Poisson cochains correspond to compatible derivations.

Experimental results

Research questions

• RQ1 How can deformations of Poisson algebras be uniformly described when both the associative and Lie products are deformed?
• RQ2 What is the role of the Markl-Remm identity in unifying associative and Lie deformations within a single nonassociative algebra structure?
• RQ3 How do Poisson-Lichnerowicz cohomology and Poisson-Hochschild cohomology relate to the full Poisson cohomology?
• RQ4 Under what conditions is a skew-symmetric or symmetric multilinear cochain a derivation for the associative or Lie product, respectively?
• RQ5 What is the precise algebraic condition under which a cochain is closed in the Poisson cohomology complex?

Key findings

• A Poisson algebra can be equivalently described as a nonassociative algebra satisfying the Markl-Remm identity, enabling a unified deformation theory.
• The Poisson cohomology complex governs general deformations, with δ_Pⁿφ = 0 if and only if δ_Cⁿφ_a = 0 and L_{1,n}φ_a = 0, where φ_a is the skew-symmetric part of φ.
• Associative deformations are parametrized by Poisson-Hochschild cohomology, and δ_Hⁿφ_s = 0 if and only if ∇ⁿφ_s = 0, meaning φ_s is a Lie n-derivation.
• The Poisson-Lichnerowicz cohomology arises as a special case when only the Lie bracket is deformed, corresponding to δ_Cⁿφ_a = 0 and L_{1,n}φ_a = 0.
• The symmetric part φ_s of a cochain satisfies δ_Hⁿφ_s = 0 if and only if ∇ⁿφ_s = 0, showing that Poisson-Hochschild cohomology coincides with Poisson cohomology for symmetric cochains.
• The full Poisson cohomology is characterized by the simultaneous vanishing of the Chevalley-Eilenberg coboundary on the skew-symmetric part and the derivation condition L_{1,n}φ_a = 0.