Sextonions, Zorn Matrices, and ${\mathbf{e}}_{7\frac{1}{2}}$

Abstract

By exploiting suitably constrained Zorn matrices, we present a new construction of the algebra of sextonions (over the algebraically closed field $\mathbb{C}$). This allows for an explicit construction, in terms of Jordan pairs, of the non-semisimple Lie algebra ${\mathbf{e}}_{7\frac{1}{2}}$, intermediate between ${\mathbf{e}}_7$ and ${\mathbf{e}}_8$, as well as of all Lie algebras occurring in the sextonionic row and column of the extended Freudenthal Magic Square.

version 2 : minor refinements, Eq. (1.2) removed, and App. A added, in which we prove that - on R - the split sextonions do not contain the divisional quaternions. 14 pages, 6 figures. version 3 : Refs. added, various refinements; matches published version on Lett. Math. Phys

References in corpus (10)

• Four-qubit entanglement from string theory
• Charge Orbits of Symmetric Special Geometries and Attractors
• Stringy Black Holes and the Geometry of Entanglement
• E_7 and the tripartite entanglement of seven qubits
• On the Black-Hole/Qubit Correspondence
• Explicit Action of E7(7) on N=8 Supergravity Fields
• The footprint of E7 in amplitudes of N=8 supergravity
• Exceptional Lie Algebras, SU(3) and Jordan Pairs Part 2: Zorn-type Representations
• The octic E8 invariant
• Oxidation of self-duality to 12 dimensions and beyond

Cited by in corpus (1)

• Jordan Algebraic Interpretation of Maximal Parabolic Subalgebras : Exceptional Lie Algebras