[Paper Review] The sextonions and $E_{7\frac 12}$

J. M. Landsberg, Laurent Manivel|ArXiv.org|Feb 10, 2004

Algebraic and Geometric Analysis23 references3 citations

TL;DR

This paper introduces the sextonions—a 6-dimensional algebra between quaternions and octonions—and uses them to fill a missing row in Freudenthal’s magic square, constructing an intermediate Lie algebra $\mathfrak{e}_{7\frac{1}{2}}$ between $\mathfrak{e}_7$ and $\mathfrak{e}_8$. The authors show that this algebra satisfies key dimension and decomposition formulas of exceptional Lie algebras, and provide geometric interpretations of adjoint varieties and quaternionic/sextonionic subalgebras in octonions.

ABSTRACT

We fill in the "hole" in the exceptional series of Lie algebras that was observed by Cvitanovic, Deligne, Cohen and deMan. More precisely, we show that the intermediate Lie algebra between $E_7$ and $E_8$ satisfies some of the decomposition and dimension formulas of the exceptional simple Lie algebras. A key role is played by the sextonions, a six dimensional algebra between the quaternions and octonions. Using the sextonions, we show simliar results hold for the rows of an expanded Freudenthal magic chart. We also obtain new interpretations of the adjoint variety of the exceptional group $G_2$.

Motivation & Objective

• To fill the 'hole' in the exceptional series of Lie algebras identified by Cvitanovic, Deligne, Cohen, and deMan, which predicted a missing dimension-6 composition algebra.
• To define and study the sextonion algebra, a non-associative 6-dimensional algebra intermediate between quaternions and octonions.
• To construct the intermediate Lie algebra $\mathfrak{e}_{7\frac{1}{2}}$ via triality applied to the sextonions, lying between $\mathfrak{e}_7$ and $\mathfrak{e}_8$.
• To extend Freudenthal’s magic square to include a new row based on the sextonions and verify that dimension and decomposition formulas of exceptional Lie algebras still hold.
• To provide new geometric interpretations of the adjoint variety of $G_2$ and the variety of quaternionic subalgebras in the octonions using sextonions.

Proposed method

• Define the sextonion algebra as a non-associative algebra over $\mathbb{C}$ of dimension 6, constructed as a subalgebra of the octonions.
• Use the triality construction from [24] to define the intermediate Lie algebra $\mathfrak{g} = \mathfrak{h} \oplus \overline{\mathfrak{g}}_1 \oplus \overline{\mathfrak{g}}_2$, where $\mathfrak{h}$ is the semisimple part of the $\mathbb{Z}$-grading of $\mathfrak{e}_8$.
• Apply the $5$-step grading of $\overline{\mathfrak{g}} = \mathfrak{e}_8$ to define the intermediate algebra $\mathfrak{g}$ as $\mathfrak{h} \oplus \overline{\mathfrak{g}}_1 \oplus \overline{\mathfrak{g}}_2$, with $\overline{\mathfrak{g}}_2 \simeq \mathbb{C}$.
• Construct the adjoint variety $X^{ad}(\mathfrak{g})$ as the closure of the orbit of highest weight vectors in $\mathbb{P}\mathfrak{g}$, parametrizing lines in the adjoint representation.
• Use the geometry of $G_{\omega}(\mathbb{H}^3, \mathbb{H}^6)$ and $G_{\omega}(\SS^3, \SS^6)$ to describe the adjoint variety of $\mathfrak{e}_{7\frac{1}{2}}$ as a $PSp(6,\SS)$-homogeneous variety.
• Analyze the closure of orbits of highest weight vectors in $\mathbb{P}V$ for preferred representations, identifying a new Severi variety with dimension 25 and one apparent double point.

Experimental results

Research questions

• RQ1 Does a 6-dimensional composition algebra exist between quaternions and octonions, and if so, what algebraic and geometric structures does it support?
• RQ2 Can the intermediate Lie algebra $\mathfrak{e}_{7\frac{1}{2}}$ be constructed via triality from the sextonions, and does it satisfy key properties of exceptional Lie algebras?
• RQ3 Do the dimension and decomposition formulas of the exceptional series extend to the new row of Freudenthal’s magic square constructed from the sextonions?
• RQ4 How can the adjoint variety of $G_2$ be geometrically interpreted using sextonionic subalgebras of the octonions?
• RQ5 What is the geometry of the closure of the orbit of a highest weight vector in the adjoint representation of $\mathfrak{e}_{7\frac{1}{2}}$, and does it yield a new Severi variety?

Key findings

• The sextonion algebra is a 6-dimensional non-associative algebra that lies between quaternions and octonions and supports a triality construction.
• The intermediate Lie algebra $\mathfrak{e}_{7\frac{1}{2}}$ is constructed as $\mathfrak{h} \oplus \overline{\mathfrak{g}}_1 \oplus \overline{\mathfrak{g}}_2$, where $\mathfrak{h} = [\overline{\mathfrak{g}}_0, \overline{\mathfrak{g}}_0]$ for $\overline{\mathfrak{g}} = \mathfrak{e}_8$, and it lies strictly between $\mathfrak{e}_7$ and $\mathfrak{e}_8$.
• The adjoint variety $X^{ad}(\mathfrak{e}_{7\frac{1}{2}})$ has dimension 25 and is isomorphic to the $PSp(6,\SS)$-orbit of the variety $G_{\omega}(\SS^3, \SS^6)$, confirming its role in the expanded magic square.
• The variety $G_{\omega}(\SS^3, \SS^6)$ has only one apparent double point, meaning its secant variety has a unique singular point, and every point outside the tangential quartic lies on a unique secant line.
• The adjoint variety $X^{ad}(\SS, \mathbb{H})$ parametrizes triples $(P, \Sigma, z)$ where $P$ is an isotropic plane in $\mathbb{C}^{12}$, $\Sigma$ is a maximal isotropic subspace containing $P$, and $z \in \mathbb{C}$, forming a 25-dimensional variety.
• A new Severi variety arises as the closure of the orbit of a highest weight vector in $\mathbb{P}\mathfrak{g}(\SS, \mathbb{H})$, which is 25-dimensional and admits a metasymplectic geometry structure in six dimensions.