[Paper Review] Operad profiles of Nijenhuis structures

Henrik Strohmayer|ArXiv.org|Sep 12, 2008

Homotopy and Cohomology in Algebraic Topology16 references3 citations

TL;DR

This paper establishes that the operad governing Nijenhuis structures and its extension for bi-Nijenhuis structures are Koszul via the PBW-basis method, proving that formal Nijenhuis and compatible Nijenhuis structures on formal manifolds correspond precisely to representations of the minimal resolutions of these operads. The key contribution is a geometric-algebraic correspondence linking operadic resolutions to differential geometric structures via Maurer-Cartan elements in graded Lie algebras of vector field-valued forms.

ABSTRACT

Recently S. Merkulov established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on the Koszul dual of a certain quadratic operad. In this paper we prove, using the PBW-basis method of E. Hoffbeck, that the operad governing Nijenhuis structures is Koszul, thereby showing that Nijenhuis structures correspond to representations of the minimal resolution of this operad. We also construct an operad such that representations of its minimal resolution in a vector space V are in one-to-one correspondence with pairs of compatible Nijenhuis structures on the formal manifold associated to V.

Motivation & Objective

• To prove the Koszulness of the operad $Πij$ governing Nijenhuis structures using Hoffbeck's PBW-basis method.
• To define and analyze the operad $βiΠij$ encoding compatible pairs of Nijenhuis structures.
• To establish a one-to-one correspondence between representations of the minimal resolution $βiΠij_{\infty}$ in a graded vector space $V$ and formal bi-Nijenhuis structures on the formal manifold associated to $V$.
• To generalize the operadic description to include formal power series solutions satisfying the Frölicher-Nijenhuis bracket condition.

Proposed method

• Apply Hoffbeck's PBW-basis method to prove Koszulness of the operads $Πij$ and $βiΠij$ by constructing compatible bases of monomials.
• Use the cobar construction on the Koszul dual cooperad to obtain the minimal resolution $Πij_{\infty}$ and $βiΠij_{\infty}$.
• Construct the dg Lie algebra ${\mathcal{L}}_{\mathcal{B}i\mathcal{N}ij}(V)$ from the total space of the operad resolution, equipped with a differential and Lie bracket.
• Define a morphism $\Phi$ from the Lie algebra of representations to the space of formal power series $\Gamma = \sum_k \Gamma_k \hslash^k$ in $\Omega^{\bullet}_V \otimes \mathcal{T}_V$.
• Verify that the Maurer-Cartan equation $[\Gamma, \Gamma]_{{}_{\text{F-N}}} = 0$ holds if and only if $\Gamma$ corresponds to a representation of $\mathcal{B}i\mathcal{N}ij_{\infty}$.
• Use the induced differential $\delta_D$ from the vector field $D$ on $V$ to describe the dg Lie algebra structure on $\tilde{{\mathfrak{g}}}_V$.

Experimental results

Research questions

• RQ1 Is the operad $\mathcal{N}ij$ governing Nijenhuis structures Koszul?
• RQ2 Can a natural operad $\mathcal{B}i\mathcal{N}ij$ be constructed to encode compatible pairs of Nijenhuis structures?
• RQ3 Does the minimal resolution $\mathcal{B}i\mathcal{N}ij_{\infty}$ classify formal bi-Nijenhuis structures on formal manifolds?
• RQ4 Is there a geometric realization of $\mathcal{B}i\mathcal{N}ij_{\infty}$-representations as formal power series satisfying the Frölicher-Nijenhuis bracket condition?
• RQ5 Can the correspondence between operadic resolutions and geometric structures be extended to graded vector spaces?

Key findings

• The operad $\mathcal{N}ij$ governing Nijenhuis structures is proven to be Koszul using the PBW-basis method, confirming that Nijenhuis structures correspond to representations of its minimal resolution $\mathcal{N}ij_{\infty}$.
• The operad $\mathcal{B}i\mathcal{N}ij$ encoding compatible Nijenhuis structures is also shown to be Koszul, enabling the construction of its minimal resolution $\mathcal{B}i\mathcal{N}ij_{\infty}$.
• There is a one-to-one correspondence between representations of $\mathcal{B}i\mathcal{N}ij_{\infty}$ in $\mathbb{R}^n$ and formal bi-Nijenhuis structures on $\mathbb{R}^n$ vanishing at the origin, proving Theorem B.
• For arbitrary graded vector spaces $V$, representations of $\mathcal{B}i\mathcal{N}ij_{\infty}$ correspond to formal power series $\Gamma = \sum_k \Gamma_k \hslash^k$ satisfying $\Gamma_k \in \Omega^{\geq k}_V \otimes \mathcal{T}_V$, $|\Gamma| = 1$, $[\Gamma, \Gamma]_{{}_{\text{F-N}}} = 0$, and $\Gamma|_0 = 0$, proving Theorem C.
• The morphism $\Phi: \mathcal{L}_{\mathcal{B}i\mathcal{N}ij}(V) \to \tilde{{\mathfrak{g}}}_V$ is an isomorphism of dg Lie algebras, establishing the geometric realization of the operadic resolution.