Related papers: E7 groups from octonionic magic square
The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group of degree 4 on this…
We introduce exceptional field theory for the group E_{7(7)}, based on a (4+56)-dimensional spacetime subject to a covariant section condition. The `internal' generalized diffeomorphisms of the coordinates in the fundamental representation…
A minimal representation of the N = 8 extended worldline supersymmetry, known as the `ultra-multiplet', is closely related to a family of supermultiplets with the same, E(8) chromotopology. We catalogue their effective symmetries and find a…
We construct the well-known decomposition of the Lie algebra $\mathfrak{e}_8$ into representations of $\mathfrak{e}_6\oplus\mathfrak{su}(3)$ using explicit matrix representations over pairs of division algebras. The minimal representation…
This paper almost classifies the maximal subgroups of $E_7(q)$ for general $q$ a power of a prime $p$. Only four potential maximal subgroups are missing: $PSL_2(7)$ (unknown for $p\neq 2,3,7$), $PSL_2(8)$ ($p=2$) and $PSL_2(9)=A_6$ ($p\neq…
By normalizing the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkahler orbifolds which satisfy the conditions to be mirror partners in…
We propose a special decomposition of the Lie $\mathfrak{su}(4)$ algebra into the direct sum of orthogonal subspaces, $\mathfrak{su}(4)=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{a}^\prime\oplus\mathfrak{t}\,,$ with…
This work provides five explicit constructions of the exceptional Lie algebra $\mathfrak{e}_8$, based on its semisimple subalgebras of maximal rank. Each of these models is graded by an abelian group, namely, $\mathbb{Z}_4$, $\mathbb{Z}_5$,…
We provide a simple parametrization for the group G2, which is analogous to the Euler parametrization for SU(2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber…
Let $G/H$ be a $p$-adic symmetric space. We compute explicitly the higher relative extension groups for all discrete series representations of $G$ in two examples: the symplectic case and the linear case. The results have immediate…
We construct the fuzzy spaces based on the three non-trivial co-adjoint orbits of the exceptional simple Lie group, $G_2$.
We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations…
We organize the homogeneous special geometries, describing as well the couplings of D=6, 5, 4 and 3 supergravities with 8 supercharges, in a small number of universality classes. This relates manifolds on which similar types of dynamical…
In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C, {E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually use the Cayley algebra $ \mathfrak{C} $. In the present article, we consider replacing…
The article is a tribute to my old mentor, collaborator and friend Murray Gell-Mann. In it I describe work by Pierre Ramond, Sung-Soo Kim and myself where we describe the N = 8 Supergravity in the light-cone formalism. We show how the…
The five exceptional simple Lie algebras over the complex number are included one within the other as $G_2 \subset F_4 \subset E_6 \subset E_7 \subset E_8$. The biggest one, $E_8$, is in many ways the most mysterious. This article surveys…
We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The…
We present a novel global E_7(7) symmetry in five-dimensional maximal supergravity as well as an E_8(8) symmetry in d=4. These symmetry groups which are known to be present after reduction to d=4 and d=3, respectively, appear as conformal…
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…
In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group $G= F_4, E_6, E_7$. We shall find involutive automorphisms of $G$ such that the connected component of the…