Octonionic structure operator and its right spectrum

Technical Summary: Octonionic Structure Operator and its Right Spectrum

Problem Statement
This paper investigates a canonical spectral problem arising from the nonassociative algebra of octonions ($\mathbb{O}$). While linear algebra over associative fields (real, complex, quaternionic) possesses well-established spectral theories, the octonionic case is subtler due to nonassociativity. Specifically, the paper addresses the right-eigenvalue problem for a natural $G_2$-equivariant operator $h$ defined on the octonionic extension $W = \mathbb{O} \otimes_{\mathbb{R}} V$, where $V$ is the standard 7-dimensional $G_2$-module (identified with the imaginary octonions $\text{Im}\,\mathbb{O}$).

The central question is to determine the set of octonions $\lambda \in \mathbb{O}$ for which there exists a non-zero vector $\hat{w} \in W$ satisfying:
$$h(\hat{w}) = \hat{w}\lambda$$
Unlike the quaternionic case, where Hermitian operators typically possess real right-eigenvalues, the nonassociativity of $\mathbb{O}$ allows for continuous families of non-real right-eigenvalues even for operators with strong symmetry properties. The paper seeks to characterize the geometry of this "right spectrum" and understand how it reflects the exceptional algebraic structure of the octonions.

Methodology
The analysis proceeds through a combination of representation theory and explicit matrix computation, leveraging the symmetry groups $G_2$ and its subgroup $SU(3)$.

1. Operator Definition: The operator $h: W \to W$ is defined intrinsically via the octonion multiplication tensor. It acts by contracting the $V$-index against the multiplication tensor and acting by left multiplication on the $\mathbb{O}$-factor. $h$ is shown to be self-adjoint and $G_2$-equivariant.
2. Real Spectrum Analysis: The ordinary real spectrum of $h$ is computed by decomposing $W$ into irreducible $G_2$-modules: $W \cong 1 \oplus 7 \oplus 7 \oplus 14 \oplus 27$. By Schur's Lemma, $h$ acts as scalars on the multiplicity-one summands ($1, 14, 27$) and as a $2 \times 2$ matrix on the isotypic component $7 \oplus 7$.
3. Reduction to Real Spectral Problems: To analyze the octonionic right-eigenvalues, the authors fix a unit imaginary octonion $\hat{u} \in \text{Im}\,\mathbb{O}$, defining a complex slice $\mathbb{C}_{\hat{u}} = \mathbb{R} \oplus \mathbb{R}\hat{u} \subset \mathbb{O}$. Any eigenvalue in this slice is written as $\lambda = \mu + Q\hat{u}$. The equation $h(\hat{w}) = \hat{w}\lambda$ is reformulated as a real eigenvalue problem for a deformed operator:
   $$H_{u,Q}(\hat{w}) = \mu \hat{w}, \quad \text{where } H_{u,Q} = h - Q R_{\hat{u}}$$
   Here, $R_{\hat{u}}$ denotes right multiplication by $\hat{u}$.
4. Symmetry Breaking and Block Decomposition: While $h$ is $G_2$-equivariant, the operator $H_{u,Q}$ is only equivariant under the stabilizer of $\hat{u}$, which is isomorphic to $SU(3)$. The space $W$ is decomposed into irreducible $SU(3)$-modules: $W \cong 1^{\oplus 4} \oplus 6^{\oplus 4} \oplus 8^{\oplus 2} \oplus 12$. The spectral problem reduces to analyzing the action of $H_{u,Q}$ on these four distinct blocks.

Key Results

1. Real Spectrum: The ordinary real spectrum of $h$ is discrete and consists of five distinct eigenvalues:
   $$\text{Spec}_{\mathbb{R}}(h) = \left\{ -7, -4, 0, \frac{1 - \sqrt{33}}{2}, \frac{1 + \sqrt{33}}{2} \right\}$$
   These correspond to the scalar actions on the $1, 14, 27$ summands and the eigenvalues of the $2 \times 2$ block on $7 \oplus 7$.

2. Octonionic Right Spectrum: The non-real right spectrum (where $Q \neq 0$) is not a discrete set but consists of continuous families lying on specific algebraic loci in the $(\mu, Q)$-plane. The analysis of the $SU(3)$-blocks yields:

   • The $12$-block and $6^{\oplus 4}$-block: These contribute no non-real right-eigenvalues for $Q \neq 0$.
   • The $1^{\oplus 4}$-block: Contributes a quartic curve defined by:
     $$\mu^4 + 6\mu^3 + (2Q^2 - 15)\mu^2 + (6Q^2 - 56)\mu + Q^4 + Q^2 = 0$$
   • The $8^{\oplus 2}$-block: Contributes a circle defined by:
     $$(\mu + 2)^2 + Q^2 = 4$$

3. Global Structure: The full right spectrum is obtained by rotating these planar loci through the $G_2$-orbit of the chosen unit imaginary direction $\hat{u}$. Since $G_2$ acts transitively on the unit sphere in $\text{Im}\,\mathbb{O}$, the spectrum forms a $G_2$-invariant hypersurface in $\mathbb{O}$. The defining equations can be expressed intrinsically using the trace $T(\lambda) = \lambda + \bar{\lambda}$ and norm $N(\lambda) = \lambda\bar{\lambda}$:

   • Quartic component: $N(\lambda)^2 + (3T(\lambda) + 1)N(\lambda) - 4T(\lambda)^2 - 28T(\lambda) = 0$
   • Circle component: $N(\lambda) + 2T(\lambda) = 0$

4. Eigenspace Dimensions: The paper details the dimensions of the eigenspaces $E_\lambda = \ker(h - R_\lambda)$. For generic points on the quartic, $\dim E_\lambda = 1$. For points on the circle, $\dim E_\lambda = 8$. At the three intersection points of the loci (one real, two non-real), the dimensions sum to 9.

Significance and Claims
The paper claims to provide a concrete example of "genuinely nonassociative spectral behavior." Unlike the quaternionic case, where noncommutativity alone does not force non-real spectra for Hermitian operators, the octonionic operator $h$ exhibits continuous families of non-real right-eigenvalues.

The significance lies in the following points:

• Symmetry-Driven Description: The spectral loci are not artifacts of a specific matrix model but emerge naturally from the branching of $G_2$ representations to $SU(3)$ upon fixing a complex slice.
• Contrast with Associative Cases: The work highlights that the octonionic right-spectrum is a continuous geometric object (a union of a quartic and a circle) rather than a discrete set of points, even for a self-adjoint, $G_2$-equivariant operator.
• Geometric Interpretation: The operator $h$ is viewed as a pointwise model for operators on octonion bundles over $G_2$-manifolds. The spectral loci represent an "alignment" between the operator $h$ and right multiplication by an octonion, encoding geometric information about the nonassociative structure that is invisible in real or complex formulations.
• Methodological Distinction: The approach differs from existing literature on small octonionic Hermitian matrices (e.g., $2 \times 2$ or $3 \times 3$) by starting with a canonical operator on a representation space and using representation theory as the primary organizing principle, rather than relying on determinant identities or specific matrix models.

The paper concludes that the right spectrum of $h$ serves as a probe of how nonassociativity enters operator theory, with the resulting spectral geometry being controlled by the interplay between $G_2$ symmetry and the choice of a complex slice.