Lie Algebra E8 Research Articles

A new division algebra representation of E6 from E8

We construct the well-known decomposition of the Lie algebra e8 into representations of e6⊕su(3) using explicit matrix representations over pairs of division algebras. The minimal representation of e6, namely the Albert algebra, is thus realized explicitly within e8, with the action given by the matrix commutator in e8, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in e8, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group E6 ⊂ E8.

A new division algebra representation of E7 from E8

We decompose the Lie algebra e8(−24) into representations of e7(−25)⊕sl(2,R) using our recent description of e8 in terms of (generalized) 3 × 3 matrices over pairs of division algebras. Freudenthal’s description of both e7 and its minimal representation are therefore realized explicitly within e8, with the action given by the (generalized) matrix commutator in e8, and with a natural parameterization using division algebras. Along the way, we show how to implement standard operations on the Albert algebra such as trace of the Jordan product, the Freudenthal product, and the determinant, all using commutators in e8.

Sporadic and Exceptional

We study the web of correspondences linking the exceptional Lie algebras E8,7,6 and the sporadic simple groups Monster, Baby and the largest Fischer group. This is done via the investigation of classical enumerative problems on del Pezzo surfaces in relation to the cusps of certain subgroups of PSL(2,R) for the relevant McKay-Thompson series in Generalized Moonshine. We also study Conway's sporadic group, as well as its association with the Horrocks-Mumford bundle.

Octions: An E8 description of the Standard Model

We interpret the elements of the exceptional Lie algebra e8(−24) as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra su(3)⊕su(2)⊕u(1), and the Lorentz Lie algebra so(3,1). Our construction relies on identifying a complex structure on spinors and then working in the enveloping algebra. The resulting model naturally contains Grand Unified Theories based on SO(10) (Georgi), SU(5) (Georgi–Glashow), and SU(4) ×SU(2) ×SU(2) (Pati–Salam). We then briefly speculate on the role of the remaining elements of e8 and propose a mechanism leading to exactly three generations of particles.

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 e8, 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.

Icosahedral Symmetry: A Review

This Review covers over 40 years of research on using the algebras of Quarternions E6;E8to model Elementary Particle physics. In particular the Binary Icosahedral group is isomorphic to theExceptional Lie algebra E8 by the MacKay correspondence. And the toric graph of E8 in Fig.2 with240 vertices on 4 binary Riemann surfaces each carrying 60 vertices, models a solution of the Ernstequation for the stationary symmetric Einstein gravitational equation. Furthermore the 15 synthemesof E8, consisting of 5 sets of 3,can be identified with algebraic representations of the nucleon,supersymmetric particles,W bosons and Dark Matter.

Parity proofs of the Kochen–Specker theorem based on the Lie algebra E8

The 240 root vectors of the Lie algebra E8 lead to a system of 120 rays in a real eight-dimensional Hilbert space that contains a large number of parity proofs of the Kochen–Specker (KS) theorem. After introducing the rays in a triacontagonal representation due to Coxeter, we present their KS diagram in the form of a ‘basis table’ showing all 2025 bases (i.e., sets of eight mutually orthogonal rays) formed by the rays. Only a few of the bases are actually listed, but simple rules are given, based on the symmetries of E8, for obtaining all the other bases from the ones shown. The basis table is an object of great interest because all the parity proofs of E8 can be exhibited as subsets of it. We show how the triacontagonal representation of E8 facilitates the identification of substructures that are more easily searched for their parity proofs. We have found hundreds of different types of parity proofs, ranging from 9 bases (or contexts) at the low end to 35 bases at the high end, and involving projectors of various ranks and multiplicities. After giving an overview of the proofs we found, we present a few concrete examples of the proofs that illustrate both their generic features as well as some of their more unusual properties. In particular, we present a proof involving 34 rays and 9 bases that appears to provide the most compact proof of the KS theorem found to date in eight-dimensions.

Icosahedral Supersymmetry and Dark Matter

The Icosahedral group has the Lie algebra E8 with a graph of 240 vertices and one real and 2 complex forms as well as a non-compact Split Form EVIII that is infinite-dimensional and shown to have 42 vertices that account for the squarks, sleptons and sneutrinos of Dark Matter.

Icosahedral Supersymmetry

The Exceptional Lie Algebra E6 used by the Author as a basis forthe Standard Model of the Elementary Particles is a subalgebra of the Lie algebra E8 which in turn is the Lie algebra of the icosahedral group by the McKay correspondence. It is possible to introduce a mass proportional toan entropy given by the the number of permutations of the elements of E6, E8 labeled by the Weyl group W. In this way the masses of the top-quark pair uu and electron are derived without any appeal to QCD and a mass of approximately 19 TeV is predicted for supersymmetric particles.

A symmetric construction of the compact real form of the Lie algebra [formula omitted

We give a new, symmetric construction of the compact real form of the complex Lie algebra E8. In this construction, the Lie product is determined using an irreducible subgroup of shape ▪ of the automorphism group of E8.

The Hamiltonian of the quantum trigonometric Calogero–Sutherland model in the exceptional algebra E8

We express the Hamiltonian of the quantum trigonometric Calogero–Sutherland model for the Lie algebra E8 and coupling constant κ by using the fundamental irreducible characters of the algebra as dynamical independent variables.

On the Microlocal Structure of the Regular Prehomogeneous Vector Space Associated with $\textit{SL}(5) \times \textit{GL}(4)$

Let p = A2 (x) AI be an irreducible representation of G = SX(5, C) x GL(4, C) on V=AC® C(= F(10) (x) F(4)). Then we have a Zariski-dense G-orbit, namely, the triplet (G, p, V) is a prehomogeneous vector space (abbrev. P.V.). All irreducible P.V.'s are completely classified (See [SK]), and the fc-functions of irreducible regular P.V.'s are already calculated (See [SKKO], [Ki], [KM], [KO], [O 2]) by using microlocal analysis (See [SKKO]) except the case of our P.V. (G, p, V) which has the most complicated microlocal structure among all the reduced irreducible regular P.V.'s (See [O 1]). The table of G-orbits in V was first given in [O 1]. However, one orbit 2 corresponding to a generic point of (SL(5) x GL(2), A2 ® A±,AC 5 (x) C) is missed in [O 1] (Remark 2.3). Later, Prof. N.Kawanaka gave the orbital decomposition of (G, p, V) by using a classification of nilpotent orbits of the exceptional Lie algebra E8 (See [Ka 1], [Ka 2]). In this paper, we shall give the detailed explanation of the original method used in [O 1] for the orbital decomposition of (G, p, V) (Proposition 2.1) and determine all good holonomic varieties (Theorem 3.4). These results are fundamental to investigate the microlocal structure of our P.V. from which we can get some information of the b-function (See [O 1]). We shall roughly explain our method. Let G' = SL(5, C) x GL(3, C), p' 2 = A2®Al, V =AC 5 (x) C. Then (G', p', V) is an irreducible P.V. and its orbital decomposition has been completed in [Ki]. By an injection GL(3, C)

The greatest mathematical paper of all time

Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.