[Paper Review] On deformation theory and graph homology

Füsun Akman, Lucian M. Ionescu+1 more|ArXiv.org|Jul 4, 2005

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper proposes a canonical solution to the Maurer-Cartan equation in deformation theory of associative algebras using an 'almost contraction' map, inspired by homotopy perturbation theory. It applies this to graph homology, showing that Moyal's star-product formula arises naturally from a merger operation in the constant Poisson structure case, and conjectures a generalization to linear Poisson structures via a homological differential.

ABSTRACT

Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an almost contraction leading to a canonical solution of the corresponding Maurer-Cartan equation is noted. This role is reminiscent of the homotopical perturbation lemma, with the infinitesimal deformation cocycle as initiator. Applied to star-products, we show how Moyal's formula can be obtained using such an almost contraction and conjecture that the merger operation provides a canonical solution at least in the case of linear Poisson structures.

Motivation & Objective

• To develop a systematic method for solving the Maurer-Cartan equation in deformation theory using an 'almost contraction' map.
• To connect deformation theory of associative algebras with graph homology through differential graded Lie algebras (DGLA).
• To show that Moyal's star-product formula emerges canonically from a merger operation in the case of constant Poisson structures.
• To conjecture that a similar merger operation provides a canonical solution for linear Poisson structures via a homological differential.
• To identify symmetry and antipodal structure in the DGLA of graphs as key to constructing such solutions.

Proposed method

• The paper uses a differential graded Lie algebra (DGLA) structure on the space of graphs to model deformation theory of associative algebras.
• It introduces an 'almost contraction' map, analogous to the homotopy perturbation lemma, to construct a canonical solution to the Maurer-Cartan equation.
• The Maurer-Cartan equation is recast as a Lie algebra equation using the graded Lie bracket, avoiding explicit use of the pre-Lie operation.
• The 'merger operation' is identified as a candidate for the almost contraction in the constant Poisson case, yielding Moyal's formula.
• For linear Poisson structures, a differential $\sigma$ is proposed as a homological operator, with $\sigma^2 = 0$, and conjectured to define a canonical solution via $Z_n = \sigma D_n$.
• The antipodal map $S(\Gamma) = (-1)^m \Gamma^t$ is used to reveal symmetry in the DGLA of graphs, supporting the existence of canonical solutions.

Experimental results

Research questions

• RQ1 Can the Maurer-Cartan equation in deformation theory be solved canonically using an almost contraction map?
• RQ2 Does the merger operation in graph homology provide a canonical solution to the star-product deformation problem in the constant Poisson structure case?
• RQ3 Is there a generalization of the merger operation to linear Poisson structures that yields a canonical star-product?
• RQ4 What role does the antipodal symmetry of the DGLA of graphs play in constructing canonical solutions?
• RQ5 How do combinatorial coefficients in the merger operation affect the construction of solutions in the general case?

Key findings

• The Maurer-Cartan equation is solved via an almost contraction map, leading to a canonical deformation solution analogous to the homotopy perturbation lemma.
• Moyal's star-product formula is derived as a canonical solution in the case of constant Poisson structures using the merger operation as the almost contraction.
• The differential $\sigma$ defined on graphs satisfies $\sigma^2 = 0$, confirming its role as a homological operator in the linear Poisson structure case.
• The antipodal map $S(\Gamma) = (-1)^m \Gamma^t$ is shown to be a Lie algebra automorphism, revealing an underlying symmetry in the DGLA of graphs.
• The conjecture $Z_n = \sigma D_n$ is proposed as a general method for constructing canonical star-products in the linear Poisson structure case.
• The proof of the Maurer-Cartan equation's solution relies on the symmetry of basis elements and the cancellation of terms via the codifferential $\delta$.