[Paper Review] The classification of orbits on certain exceptional Jordan algebra under the automorphism group

Akihiro Nishio|arXiv (Cornell University)|Nov 3, 2010

Advanced Topics in Algebra10 references3 citations

TL;DR

This paper classifies the orbits of the exceptional Lie group $\mathrm{F}_{4(-20)}$ acting on the real exceptional Jordan algebra $\mathcal{J}^1$, providing a complete decomposition of $\mathcal{J}^1$ into $\mathrm{F}_{4(-20)}$-orbits based on characteristic roots and minimal subspaces. The key contribution is a complete orbit decomposition and the explicit determination of the Lie group structure of stabilizers for each orbit, including stabilizers isomorphic to $\mathrm{Spin}(7)\ltimes\mathrm{H}_{\mathrm{Im}\mathbf{O},\mathbf{O}}$ and $\mathrm{G}_2\ltimes\mathrm{H}_{\mathrm{Im}\mathbf{O},\mathrm{Im}\mathbf{O}}$. The classification is based on the characteristic polynomial and the minimal subspace $V_X$ generated by $E$ and $X$, with canonical forms given for all orbit types.

ABSTRACT

Let $\mathcal{J}^1$ be the real form of complex simple Jordan algebra with the automorphism group $G$ of type $F_{4(-20)}$. Explicitly, we give the orbit decomposition of $\mathcal{J}^1$ under the action of $G$ and determine the Lie group structure of stabilizer for each $G$-orbit on $\mathcal{J}^1$.

Motivation & Objective

• To classify the $\mathrm{F}_{4(-20)}$-orbits on the real exceptional Jordan algebra $\mathcal{J}^1$ under the action of its automorphism group.
• To determine the Lie group structure of the stabilizer subgroup for each orbit in $\mathcal{J}^1$.
• To provide explicit canonical forms for each orbit type using characteristic roots and the minimal subspace $V_X$ generated by $E$ and $X$.
• To extend previous orbit classifications for $\mathrm{F}_{4}^{\mathbb{C}}$ and $\mathrm{F}_{4(4)}$ to the split real form $\mathrm{F}_{4(-20)}$.
• To establish a complete and explicit orbit decomposition based on the spectral and algebraic structure of elements in $\mathcal{J}^1$.

Proposed method

• The orbit decomposition is based on the characteristic polynomial $\Phi_X(\lambda)$ of $X \in \mathcal{J}^1$, with classification depending on the nature of its roots: three distinct real roots, one real and two complex conjugate roots, or multiple roots.
• The minimal subspace $V_X$ is defined as the smallest $\mathrm{F}_{4(-20)}$-invariant subspace containing $E$ and $X$, closed under the cross product, and used to analyze orbit intersections with standard orbits like $\mathcal{H}(\mathbf{O})$, $\mathcal{H}'(\mathbf{O})$, and $\mathcal{N}_1^{\pm}(\mathbf{O})$.
• Canonical forms are constructed using diagonal matrices and specific elements $P^+, P^-, Q^+(1)$, with the action of $\mathrm{F}_{4(-20)}$ used to transform any $X$ into one of these standard forms.
• The stabilizer structure is determined via conjugation and automorphism techniques, using known isomorphisms such as $\tilde{\sigma}((\mathrm{F}_{4(-20)})_Z) \cong (\mathrm{F}_{4(-20)})_{P^-}$ and $\mathrm{F}_{4(-20)}$-invariance of $E$.
• Propositions 8.8, 8.10, 8.11, and 8.12 establish stabilizer isomorphisms using known group structures: $\mathrm{Spin}(7)\ltimes\mathrm{H}_{\mathrm{Im}\mathbf{O},\mathbf{O}}$ and $\mathrm{G}_2\ltimes\mathrm{H}_{\mathrm{Im}\mathbf{O},\mathrm{Im}\mathbf{O}}$.
• The classification relies on the invariance of trace, inner product, determinant, and characteristic roots under $\mathrm{F}_{4(-20)}$-action, ensuring orbit invariance.

Experimental results

Research questions

• RQ1 How are the $\mathrm{F}_{4(-20)}$-orbits on the real exceptional Jordan algebra $\mathcal{J}^1$ classified based on the characteristic roots of elements?
• RQ2 What is the Lie group structure of the stabilizer subgroup for each $\mathrm{F}_{4(-20)}$-orbit in $\mathcal{J}^1$?
• RQ3 How do the minimal subspaces $V_X$ generated by $E$ and $X$ help in determining the orbit type of $X$?
• RQ4 What canonical forms represent each orbit type under the $\mathrm{F}_{4(-20)}$-action?
• RQ5 How do the stabilizers of $P^+$, $P^-$, and $Q^+(1)$ relate to known Lie groups such as $\mathrm{Spin}(7)\ltimes\mathrm{H}_{\mathrm{Im}\mathbf{O},\mathbf{O}}$ and $\mathrm{G}_2\ltimes\mathrm{H}_{\mathrm{Im}\mathbf{O},\mathrm{Im}\mathbf{O}}$?

Key findings

• The $\mathrm{F}_{4(-20)}$-orbits on $\mathcal{J}^1$ are completely classified into 12 canonical types based on the spectrum of $X$ and the structure of $V_X$.
• For elements with three distinct real characteristic roots $\lambda_1 > \lambda_2 > \lambda_3$, the orbit is determined by which $E_{X,\lambda_i}$ lies in $\mathcal{H}(\mathbf{O})$, with canonical forms $\mathrm{diag}(\lambda_i, \lambda_{i+1}, \lambda_{i+2})$.
• For elements with one real root and a complex conjugate pair $p \pm iq$, $q > 0$, the canonical form is $\mathrm{diag}(p,p,\lambda_1) + F_3^1(q)$, corresponding to orbit $\mathcal{N}_1^+(\mathbf{O})$ or $\mathcal{N}_1^-(\mathbf{O})$.
• When $X$ has a single real root $\lambda_1$ of multiplicity one and $\lambda_2$ of multiplicity two, the orbit depends on whether $W_{X,\lambda_1} = 0$ or lies in $\mathcal{N}_1^{\pm}(\mathbf{O})$, yielding canonical forms $\mathrm{diag}(\lambda_2,\lambda_2,\lambda_1)$, $\mathrm{diag}(\lambda_2,\lambda_2,\lambda_1) + P^+$, or $\mathrm{diag}(\lambda_2,\lambda_2,\lambda_1) + P^-$.
• For elements with a triple root, the orbit is determined by the traceless part $p(X)$: if $p(X) = 0$, the orbit is $3^{-1}\mathrm{tr}(X)E$; if $p(X) \in \mathcal{N}_1^{\pm}(\mathbf{O})$, the form is $3^{-1}\mathrm{tr}(X)E \pm P^\pm$; if $p(X) \in \mathcal{N}_2(\mathbf{O})$, the form is $3^{-1}\mathrm{tr}(X)E + Q^+(1)$.
• The stabilizer of $P^-$ is isomorphic to $\mathrm{Spin}(7)\ltimes\mathrm{H}_{\mathrm{Im}\mathbf{O},\mathbf{O}}$, and the stabilizer of $Q^+(1)$ is isomorphic to $\mathrm{G}_2\ltimes\mathrm{H}_{\mathrm{Im}\mathbf{O},\mathrm{Im}\mathbf{O}}$, confirming known group structures in the context of $\mathrm{F}_{4(-20)}$.