Magic Squares of Lie Algebras

Sextonions and the magic square

Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an additional row and column. The extra row and column in the magic square corresponds to the sextonions. This is a six dimensional subalgebra of the split octonions which contains the split quaternions.

The extended Freudenthal Magic Square and Jordan algebras

The Lie superalgebras in the extended Freudenthal Magic Square in characteristic 3 are shown to be related to some known simple Lie superalgebras, specific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jordan algebras.

Magic squares and matrix models of Lie algebras

This paper is concerned with the description of exceptional simple Lie algebras as octonionic analogues of the classical matrix Lie algebras. We review the Tits-Freudenthal construction of the magic square, which includes the exceptional Lie algebras as the octonionic case of a construction in terms of a Jordan algebra of hermitian 3x3 matrices (Tits) or various plane and other geometries (Freudenthal). We present alternative constructions of the magic square which explain its symmetry, and show explicitly how the use of split composition algebras leads to analogues of the matrix Lie algebras …

An extended Freudenthal Magic Square in characteristic 3

Freudenthal's Magic Square, which in characteristic 0 contains the exceptional Lie algebras other than G2, is extended over fields of characteristic 3, through the use of symmetric composition superalgebras, to a larger square that contains both Lie algebras and superalgebras. With one exception, the simple Lie superalgebras that appear have no counterpart in characteristic 0.

The Magic Square and Symmetric Compositions II

The construction of Freudenthal's Magic Square, which contains the exceptional simple Lie algebras, in terms of symmetric composition algebras is further developed here. The para-Hurwitz algebras, which form a subclass of the symmetric composition algebras, will be defined, in the split case, in terms of the natural two dimensional module for the simple Lie algebra sl(2). As a consequence, it will be shown how all the Lie algebras in Freudenthal's Magic Square can be constructed, in a unified way, using copies of sl(2) and of its natural module.

A Kind of Magic

We introduce the extended Freudenthal-Rosenfeld-Tits magic square based on six algebras: the reals $\mathbb{R}$, complexes $\mathbb{C}$, ternions $\mathbb{T}$, quaternions $\mathbb{H}$, sextonions $\mathbb{S}$ and octonions $\mathbb{O}$. The ternionic and sextonionic rows/columns of the magic square yield non-reductive Lie algebras, including $\mathfrak{e}_{7\scriptscriptstyle{\frac{1}{2}}}$. It is demonstrated that the algebras of the extended magic square appear quite naturally as the symmetries of supergravity Lagrangians. The sextonionic row (for appropriate choices of real forms) gives th…

反射与旋转的幻方

The magic square of reflections and rotations

本文基于 Coxeter 的工作建立秩二复反射群与 $O(3)$ 实反射群的双射，并在 Clifford 代数与 (s)pin 群框架下解释并探索小维度的反射旋转幻方。

The Mile High Magic Pyramid

Using a unified formulation of $\mathcal{N} = 1, 2, 4, 8$, super Yang-Mills theories in $D = 3$ spacetime dimensions with fields valued respectively in $\mathbb{R, C, H, O}$, it was shown that tensoring left and right multiplets yields a Freudenthal magic square of $D = 3$ supergravities. When tied in with the more familiar $\mathbb{R, C, H, O}$ description of super Yang-Mills in $D = 3, 4, 6, 10$ this results in a magic pyramid of supergravities: the known $4 \times 4$ magic square at the base in $D=3$, a $3\times 3$ square in $D=4$, a $2 \times 2$ square in $D=6$ and Type II supergravity at …

三重性与汉斯·弗里德里希的魔方

Triality and the Magic Square of Hans Freudenthal

本文通过张量代数研究实三重性结构，构建了双三重性算子的李代数，证明了雅可比恒等式，并将结果与魔方中的相应条目统一。还探讨了由此产生的不变双线性形式和克利夫顿理论结构。

Magic Star and Exceptional Periodicity: an approach to Quantum Gravity

We present a periodic infinite chain of finite generalisations of the exceptional structures, including the exceptional Lie algebra $\mathbf{e_8}$, the exceptional Jordan algebra (and pair) and the octonions. We will also argue on the nature of space-time and indicate how these algebraic structures may inspire a new way of going beyond the current knowledge of fundamental physics.

A magic pyramid of supergravities

By formulating N = 1, 2, 4, 8, D = 3, Yang-Mills with a single Lagrangian and single set of transformation rules, but with fields valued respectively in R,C,H,O, it was recently shown that tensoring left and right multiplets yields a Freudenthal-Rosenfeld-Tits magic square of D = 3 supergravities. This was subsequently tied in with the more familiar R,C,H,O description of spacetime to give a unified division-algebraic description of extended super Yang-Mills in D = 3, 4, 6, 10. Here, these constructions are brought together resulting in a magic pyramid of supergravities. The base of the pyrami…

A magic approach to octonionic Rosenfeld spaces

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfe…

On the varieties of the second row of the split Freudenthal-Tits Magic Square

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type $\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone's approach to finite quadric Veronesean varieties. This approach takes projective properties…

On Superalgebras of Matrices with Symmetry Properties

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices give rise to similar superalgebra structures, pointing to novel symmetry types in their complementary parts. In particular, this provides a unifying framework for the composite `most perfect square' symmetry and the related class of `reversible squares'; moreover, the semi-magic square algebra is identified as part of a 2-gradation of the general…

On the real forms of the exceptional Lie algebra $ \mathfrak{e}_6$ and their Satake diagrams

Satake diagrams of the real forms $ \mathfrak{e}_{6,-26}$, $ \mathfrak{e}_{6,-14}$ and $ \mathfrak{e}_{6,2}$ are carefully developed. The first real form is constructed with an Albert algebra and the other ones by using the two paraoctonion algebras and certain symmetric construction of the magic square.