[Paper Review] Symplectic operad geometry and graph homology

Swapneel Mahajan|ArXiv.org|Nov 29, 2002

Homotopy and Cohomology in Algebraic Topology23 references3 citations

TL;DR

This paper formulates Kontsevich's theorem relating Lie algebra homology to graph homology in the general setting of reversible operads, using a pictorial calculus of cuttings and matings. It constructs a Lie algebra QA∞ and a graph complex (QG, ∂E) from a mated species Q, proving their homologies are isomorphic, thereby unifying and generalizing Kontsevich's original results for commutative, associative, and Lie algebras via symplectic operad geometry.

ABSTRACT

A theorem of Kontsevich relates the homology of certain infinite dimensional Lie algebras to graph homology. We formulate this theorem using the language of reversible operads and mated species. All ideas are explained using a pictorial calculus of cuttings and matings. The Lie algebras are constructed as Hamiltonian functions on a symplectic operad manifold. And graph complexes are defined for any mated species. The general formulation gives us many examples including a graph homology for groups. We also speculate on the role of deformation theory for operads in this setting.

Motivation & Objective

• To generalize Kontsevich's theorem relating Lie algebra homology to graph homology beyond the three classical cases (commutative, associative, Lie).
• To formulate this connection in the natural setting of reversible operads, using the language of species and mated species.
• To construct a Lie algebra QA∞ as Hamiltonian functions on a symplectic operad manifold and define a corresponding graph complex (QG, ∂E).
• To prove that the homology of QA∞ is isomorphic to the homology of the graph complex (QG, ∂E), extending Kontsevich's result to a broader class of operads.

Proposed method

• Introduces the Mating Functor from reversible operads to species, mapping a reversible operad P to a species Q, which serves as the foundation for constructing QA∞ and (QG, ∂E).
• Uses a pictorial calculus of cuttings and matings to define the Lie algebra QA∞ as a direct limit of Lie algebras QAn.
• Defines the graph complex (QG, ∂E) using a boundary map ∂E that is a derivation with respect to the product µ, and introduces a secondary bracket [ , ] = ∂Hµ − µ∂H to measure failure of ∂H to commute with µ.
• Applies deformation theory by introducing a gauge-equivalent deformation of the trivial deformation of G[[t]], using a map D(n) = A−1 ◦ M(n) to define D = D0 + tD1 + t²D2 + ..., where D0 is the identity.
• Uses the deformation to show that ∂H and the bracket [ , ] induce the zero map on homology, with D1 and µ1 as chain homotopies.
• Establishes that ∂H and [ , ] are trivial on homology by showing ∂1 = −∂H and [ , ] = ∂Eµ1 − µ1∂E, relying on the adjoint property of the pairing M(n) and the commutativity of diagrams in the deformation.

Experimental results

Research questions

• RQ1 How can Kontsevich’s theorem relating Lie algebra homology to graph homology be generalized beyond the three classical cases (commutative, associative, Lie)?
• RQ2 What is the role of reversible operads and mated species in unifying the construction of Lie algebras and graph complexes?
• RQ3 How does the pictorial calculus of cuttings and matings facilitate the construction of the Lie algebra QA∞ and the graph complex (QG, ∂E)?
• RQ4 In what way does deformation theory provide a conceptual framework for proving that ∂H and the bracket [ , ] are trivial on homology?
• RQ5 Can the structure of the graph complex (QG, ∂E) be extended to include other algebraic objects, such as groups, via the mated species construction?

Key findings

• The Lie algebra QA∞ is constructed as a direct limit of Lie algebras QAn, with its homology carrying a Hopf algebra structure.
• The graph complex (QG, ∂E) is defined using a boundary map ∂E that is a derivation with respect to the product µ, ensuring compatibility with the algebraic structure.
• The secondary bracket [ , ] = ∂Hµ − µ∂H measures the failure of ∂H to commute with µ, and it induces the zero map on homology.
• The map ∂H induces the zero map on homology, with D1 providing a chain homotopy, as shown via deformation theory and gauge equivalence.
• The product µ1, derived from the deformation map D(n), satisfies [ , ] = ∂Eµ1 − µ1∂E, proving the bracket is trivial on homology.
• The coproduct ∆ on G is preserved under ∂E but not under ∂H, and the cobracket θ = ∂H∆ − ∆∂H also induces the zero map on homology.