Symplectic Structure of 𝒮⁺

• Symplectic Structure of 𝒮⁺ is a framework combining invariant theory and canonical metrics, underpinning radiative phase space and BMS symmetries.
• The structure employs algebraic decompositions using Young diagrams and p-stable bases from chord diagram methods to regulate invariant operations.
• It reveals deep arithmetic and cohomological interactions through Galois obstructions and splitting sequences that link topology with algebraic constraints.

The symplectic structure of $\mathscr{I}^+$ (future null infinity) encompasses a diverse landscape of algebraic, geometric, and physical manifestations. In mathematics, it is analyzed through the invariant theory of symplectic Lie algebras, the geometry of stratified spaces, and canonical connections in parabolic conformally symplectic structures. In mathematical physics, particularly in general relativity and scattering theory, $\mathscr{I}^+$ is the universal structure supporting radiative phase space, BMS symmetries, and covariant charges, often formalized via symplectic or twistorial frameworks. The technical underpinnings span canonical metrics, cohomological definitions, group actions, and complex symplectic involutions.

1. Symplectic Invariant Structure: Metrics and Decomposition

In the context of free Lie algebras generated by the rational homology group $H_\mathbb{Q}$ of a closed oriented surface $\Sigma_g$ of genus $g$, the Sp-invariant part $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$ is canonically metrized. Each tensor power $(H_\mathbb{Q}^{\otimes 2k})^{\mathrm{Sp}}$ inherits an inner product from the intersection pairing $\mu^{\otimes 2k}$. This leads to an explicit orthogonal decomposition: $$(H_\mathbb{Q}^{\otimes 2k})^{\mathrm{Sp}} \cong \bigoplus_{|\lambda|=k,\, h(\lambda) \leq g} U_\lambda,$$ where $U_\lambda$ corresponds to a Young diagram $\lambda$ (with $k$ boxes), and each basis is further labeled by the eigenvalue

$$\mu_\lambda = \prod_{b \in \lambda} (2g - 2s_b + t_b),$$

where $s_b$ (columns to the left) and $t_b$ (rows above) index boxes in $\lambda$. For the algebra $\mathfrak{h}_{g,1}$, which is a subspace of $(H_\mathbb{Q}^{\otimes (2k+2)})^{\mathrm{Sp}}$, one writes

$$\mathfrak{h}_{g,1}(2k)^{\mathrm{Sp}} \cong \bigoplus_{|\lambda|=k+1,\, h(\lambda)\leq g} H_\lambda,$$

leading to a canonical decomposition of any Sp-invariant element into $\lambda$-coordinates with respect to the canonical metric (Morita et al., 2014). This granular control is crucial for analyzing all higher algebraic structures derived from free Lie algebras with respect to the mapping class group.

2. Constraints on Brackets and Invariant Operations

The algebraic bracket $[\, , \, ]$ in $\mathfrak{h}_{g,1}$ is highly constrained by the Sp-invariant structure. Decomposing into GL-isotypical components, with $$\widetilde{H}_\lambda \subset H_\mathbb{Q}^{\otimes (k+2)}$$ for Young diagrams $\lambda$, the bracket

$$[\widetilde{H}_\lambda,\, \widetilde{H}_\mu] \subset \mathfrak{h}_{g,1}(k+\ell),$$

lies only in irreducible components indexed by diagrams $\nu$ (with $k+\ell+2$ boxes) that satisfy:

• The tensor product $\lambda_{\mathrm{GL}}\otimes \mu_{\mathrm{GL}}$ and $$(\wedge^2 H_\mathbb{Q} \otimes \nu^\delta_{\mathrm{GL}})$$ share a common GL-summand.

This imposes a "height" constraint: only $\nu$ with number of rows between $\max\{h(\lambda), h(\mu)\} - 2$ and $h(\lambda)+h(\mu)$ appear. Thus most potential bracket outputs are algebraically forbidden (Morita et al., 2014). The structure of $\mathscr{I}^+$ in this algebraic context is sharply controlled by these invariant constraints, greatly restricting possible deformations and extensions.

3. Relations Among $\mathfrak{h}_{g,1}$, $\mathfrak{h}_{g,*}$, and $\mathfrak{h}_g$

The Sp-invariant Lie subalgebras are related via short exact sequences over $\mathbb{Z}$ or $\mathbb{Q}$: $$0 \to \mathfrak{j}^Z_{g,1} \to \mathfrak{h}^Z_{g,1} \to \mathfrak{h}^Z_{g,*} \to 0,$$

$$0 \to \mathcal{L}^Z_g \to \mathfrak{h}^Z_{g,*} \to \mathfrak{h}^Z_g \to 0.$$

Here, $\mathfrak{j}_{g,1}$ is the kernel (the "boundary" part) associated with projection from a surface with boundary, and $\mathcal{L}_g$ is the Lie algebra from the lower central series of $\pi_1(\Sigma_g)$. Labute’s theorem enables GL-decomposition and an explicit direct sum: $$\mathfrak{h}_{g,1}(k) \cong \mathfrak{j}_{g,1}(k) \oplus \mathcal{L}_g(k) \oplus \mathfrak{h}_g(k).$$ This splitting is critical for distinguishing the boundary phenomenon, the Lie algebraic content of the fundamental group, and the true symplectic derivations without boundary (Morita et al., 2014). Degree-wise computation delineates the contributions from each part to $\mathscr{I}^+$.

4. Construction Techniques: Linear Chord Diagrams and Stability

The construction of explicit elements in $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$ leverages linear chord diagrams. There is a canonical map

$$\Phi: \mathbb{Q}\mathcal{D}^{\ell}(2k+2) \to (H_\mathbb{Q}^{\otimes (2k+2)})^{\mathrm{Sp}},$$

and one employs symmetric group operators (such as $$S_{2k+2}'\circ \sigma_{2k+2}\circ p_{2k+1}'\circ \sigma_{2k+2}^{-1}$$) to produce elements in $\mathfrak{h}_{g,1}(2k)^{\mathrm{Sp}}$. Through carefully crafted bases $\{C_{\lambda}^i\}$ for diagram subspaces $F_\lambda$, mapped to $H_\lambda$, one builds a "p-stable" basis exhibiting naturality under genus projection.

Normalizing these bases via the corresponding eigenvalues $\mu_\lambda$ leads to the notion of "weighted stability," rendering the description independent of genus after rescaling. The Sp-invariant part of the ideal $\mathfrak{j}_{g,1}$ is p-stable, while the Johnson image is only weighted stable after normalization. This separation reflects fundamental differences in the geometric content encoded at $\mathscr{I}^+$ (Morita et al., 2014).

5. Galois Obstructions and Arithmetic Implications

A distinct arithmetic phenomenon—Galois obstructions—emerges prominently in the genus-one case. Specifically, in degree six, the kernel of the Enomoto–Satoh map

$$\mathrm{Ker}~\mathrm{ES}_6/\mathrm{Im}~\tau_{g,1}(6)$$

contains one extra Sp-invariant dimension absent from the Johnson image: $$\dim((\mathrm{Ker}~\mathrm{ES}_6/\mathrm{Im}~\tau_{g,1}(6))^{\mathrm{Sp}}) = 1.$$ This extra "obstruction" is identified as originating in the arithmetic (Galois theoretic) structure, corresponding to nontrivial contributions to the mapping class group that are invisible to the classical Johnson homomorphism. These elements normalize the Johnson image within $\mathfrak{h}_{g,1}(6)^{\mathrm{Sp}}$ and embody the interplay between number theory and topological invariants at $\mathscr{I}^+$ (Morita et al., 2014).

6. Cohomological and Geometric Interpretations

The symplectic structure at $\mathscr{I}^+$, in light of these algebraic results, is tightly linked to representation-theoretic and geometric data:

• The canonical metric (defined via $\mu^{\otimes 2k}$) provides a nondegenerate inner product underpinning all invariant structures.
• Orthogonal direct sum decompositions indexed by Young diagrams yield fine-grained control over the Sp-invariant tensors and allow the construction of explicit, well-behaved bases.
• The structure of brackets and the presence of arithmetic obstructions manifest in cohomological invariants that are visible in both direct computations and in the splitting of the relevant Lie algebra sequences.

Synthesis of these approaches clarifies that the symplectic structure of $\mathscr{I}^+$ is not ad hoc but follows from deep algebraic constraints, constructive techniques rooted in the topology of surfaces, and arithmetic phenomena mediated by the absolute Galois group.

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Summary Table: Key Structural Components

Algebraic Feature	Description	Role at $\mathscr{I}^+$
Canonical metric ($\mu$)	Induces orthogonal decomposition indexed by Young diagrams	Controls invariant splitting
Bracket constraints	Bounds on summands in $[\widetilde{H}_\lambda, \widetilde{H}_\mu]$ via tensor product conditions	Restricts allowed algebraic operations
Short exact sequences	Relate $\mathfrak{h}_{g,1}$, $\mathfrak{j}_{g,1}$, $\mathcal{L}_g$, $\mathfrak{h}_g$	Encodes boundary, fundamental group, and derivation content
Chord diagram method	Explicit construction of Sp-invariants and bases with (weighted) stability properties	Enables well-structured basis construction
Galois obstructions	One-dimensional extra Sp-invariant summands in certain degrees, detected in genus one	Highlights deep arithmetic-topological interaction

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The symplectic structure at $\mathscr{I}^+$ is thus an overview of canonical invariant theory, explicit constructive techniques, and arithmetic-topological phenomena—manifested both in the fine structure of invariant Lie subalgebras and in the broader geometric and physical interpretations relevant to the structure of null infinity.