Related papers: Symmetric Spaces of Exceptional Groups
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…
Magical supergravities are a very special class of supergravity theories whose symmetries and matter content in various dimensions correspond to symmetries and underlying algebraic structures of the remarkable geometries of the Magic Square…
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…
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 $\mathbb{O}$.…
This article contains the last part of the mini-course `Spaces: a perspective view' delivered at the IFWGP2012. Here I deal with the part of the mini-course which centers on the classification questions associated to the simple real Lie…
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…
Quarks and leptons charges and interactions are derived from gauge theories associated with symmetries. Their space-time labels come from representations of the non-compact algebra of Special Relativity. Common to these descriptions are the…
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…
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…
A unified treatment of the $2 \times 2$ analog of the Freudenthal-Tits magic square of Lie groups is given, providing an explicit representation in terms of matrix groups over composition algebras.
We consider four supergravities with 16+16, 32+32, 64+64, 128+128 degrees of freedom displaying some curious properties: (1) They exhibit minimal supersymmetry (N=1, 2, 2, 1) but maximal rank (r=7, 6, 4, 0) of the scalar coset in D=4, 5, 7,…
We construct maximal supergravity in five-dimensions by 'oxidizing' the four-dimensional $\mathcal{N}=8$ theory. The relevant symmetries, the unitary symplectic group $USp(8)$ and the exceptional group $E_6$, are both presented in…
Eleven-dimensional supergravity reveals large exceptional symmetries upon reduction, in accordance with the U-duality groups of M-theory, but their higher-dimensional geometric origin has remained a mystery. In this letter, we show that…
We construct the 12-dimensional exceptional field theory associated to the group $\mathrm{SL}(2) \times \mathbb{R}^+$ . Demanding the closure of the algebra of local symmetries leads to a constraint, known as the section condition, that…
We are answering the question why 4-dimensional space has the metric 1+3 by making a general argument from a certain type of equations of motion linear in momentum for any spin (except spin zero) in any even dimension d. All known free…
We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier-Saito theorem for solvable Lie groups. We then give a…
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…
We describe a remarkable rank fourtenn matrix factorization of the octic Spin(14)-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor…
We investigate nonintegrable Riemannian geometries modelled after certain symmetric spaces related to the Freudenthal-Tits Magic Square. The collection of four such structures found by Nurowski is extended by further eight. A focus is given…
In six-dimensional F-theory/heterotic string theory, half-hypermultiplets arise only when they correspond to particular quaternionic K\"ahler symmetric spaces, which are mostly associated with the Freudenthal-Tits magic square. Motivated by…