Exceptional Lie Research Articles

EXAMPLES OF FREE ACTIONS ON PRODUCTS OF SPHERES

We construct a non-abelian extension $\Gamma$ of $S^1$ by $\cy 3 \times \cy 3$, and prove that $\Gamma$ acts freely and smoothly on $S^{5} \times S^{5}$. This gives new actions on $S^{5} \times S^{5}$ for an infinite family $\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_2$ admits a free smooth action on $S^{11}\times S^{11}$. This gives new actions on $S^{11}\times S^{11}$ for an infinite family $\cE $ of finite groups. We explain the significance of these families $\cP $, $\cE $ for the general existence problem, and correct some mistakes in the literature.

A Geometric Construction of the Exceptional Lie Algebras F 4 and E 8

We present a geometric construction of the exceptional Lie algebras F4 and E8 starting from the round 8- and 15-spheres, respectively, inspired by the construction of the Killing superalgebra of a supersymmetric supergravity background. (There is no supergravity in the paper.)

Knot wormholes and the dimensional invariant of exceptional Lie groups and Stein space hierarchies

The present short note points out a most interesting and quite unexpected connection between the number of distinct knot as a function of their crossing number and exceptional Lie groups and Stein space hierarchies. It is found that the crossing number 7 plays the role of threshold similar to 4 and 5 in E-infinity theory and for the 11 crossing the number of distinct knots is very close to 4 α ¯ 0 + 1 = 548 + 1 = 549 , where α ¯ 0 = 137 is the inverse integer electromagnetic fine structure constant. This is particularly intriguing in view of a similar relation pertinent to the 17 two and three Stein spaces where the total dimension is ∑ 1 17 Stein = 5 α ¯ 0 + 1 = 685 + 1 = 686 , as well as the sum of the eight exceptional Lie symmetry groups ∑ i = 1 8 | E i | = 4 α ¯ 0 = 548 . The slight discrepancy of one is explained in both cases by the inclusion of El Naschie’s transfinite corrections leading to ∑ i = 1 8 | E i | = ( 4 ) ( 137 + k 0 ) = 548.328157 and ∑ i = 1 17 Stein = ( 5 ) ( 137 + k 0 ) = 685.41097 , where k o = ϕ 5(1 − φ 5) and φ = 5 - 1 / 2 .

Kac–Moody exceptional E12 from simplictic tiling

We give various derivations for the order of a new non classical exceptional Lie group E12. We start from the simplest polyhedra of ordinary three dimensional space and arrive at the exact integer value ∣E12∣ = 685. Subsequently we show that a corresponding infinite dimensional but hierarchal KAC–Moody algebra called 4D fusion algebra leads to an exact transfinite dimension equal to Dim E 12 = ( 5 ) ( α ¯ o ) = 685.410197 , where α ¯ o = 137.082039325 is the E-infinity electromagnetic fine structure constant.

$S_4$-symmetry on the Tits construction of exceptional Lie algebras and superalgebras

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 $S_4$ on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two different types of structurable algebras. Some models of exceptional Lie superalgebras are discussed too.

$S_4$-symmetry on the tits construction of exceptional Lie algebras and superalgerbras

$S_4$-symmetry on the tits construction of exceptional Lie algebras and superalgerbras

Quasi exceptional E12 Lie symmetry group with 685 dimensions, KAC-Moody algebra and E-infinity Cantorian spacetime

The short note gives a derivation for a new E12 exceptional Lie group corresponding to affine KAC-Moody algebra. We derive the dimension of the group by intersectionally embedding the intrinsic dimension of E8 namely D( E8) = 57 into the 12 spacetime dimensions of F theory and finding that Dim E12 = D( E8) ( DF) + 1 = (57)(12) + 1 = 685.

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set Ω. A base for G is a subset B ⊆ Ω with pointwise stabilizer in G that is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ⩽ 6 if G is an almost simple group of exceptional Lie type and Ω is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) ⩽ 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios.

Deriving the largest expected number of elementary particles in the standard model from the maximal compact subgroup H of the exceptional Lie group [formula omitted

The maximal number of elementary particles which could be expected to be found within a modestly extended energy scale of the standard model was found using various methods to be N = 69. In particular using E-infinity theory the present Author found the exact transfinite expectation value to be 〈 N 〉 = α ¯ o / 2 ≅ 69 where α ¯ o = 137.082039325 is the exact inverse fine structure constant. In the present work we show among other things how to derive the exact integer value 69 from the exceptional Lie symmetry groups hierarchy. It is found that the relevant number is given by dim H = 69 where H is the maximal compact subspace of E 7 ( - 5 ) so that N = dim H = 69 while dim E 7 = 133.

Yang–Mills instanton via exceptional Lie symmetry groups and E-infinity

The self-dual instanton solution of an Euclidean Yang–Mills theory is expressed in terms of exceptional Lie groups and E-infinity theory. Highly interesting analogies between the instanton of quantum field in 3 + 1-dimensions and K3 Kähler of superstrings are also discussed.

Using Witten’s five Brane theory and the holographic principle to derive the value of the electromagnetic fine structure constant [formula omitted

Using the 528 states of Witten’s five Brane theory in conjunction with the holographic principles we give various derivations for the inverse electromagnetic fine structure constant. Relations to the exceptional Lie group F 4 used to model a gravity-weak field are also considered.

A simple direct connection between superstrings and E8 unification

This work is necessarily a short letter with a calculation which can be done virtually on the back of an envelope. The result although simple is far reaching. It is shown that embedding E8 in the bosonic strings spacetime leads to a theory consistent with that based on summing over an exceptional Lie symmetry group hierarchy with the invariant total dimension D = 4 α ¯ 0 = ( 4 ) ( 137 ) = 548 .

E-eight exceptional Lie groups, Fibonacci lattices and the standard model

This short paper is intended to disclose a most interesting connection between the roots system of the exceptional Lie groups family and the standard model. First an accurate correspondence is found between the roots number of the different groups and the various sections of the mass-spectrum of the standard model and beyond. Second it is shown that the exact gauging of the renormalization group equation is not a logarithmic but rather a fractal scaling related to the geometrical mean of exceptional Lie group family or a Fibonacci line describing the extended standard model. This family may be approximated by ( α ¯ 0 ) ( φ 3 ) = 32.3606399 as an average where α ¯ 0 = 137.082033989 is the E-infinity exact value for the inverse fine structure constant and φ = ( 5 - 1 ) / 2 .

Quantum Symmetric Pairs and the Reflection Equation

It is shown that central elements in G. Letzter’s quantum group analogs of symmetric pairs lead to solutions of the reflection equation. This clarifies the relation between Letzter’s approach to quantum symmetric pairs and the approach taken by M. Noumi, T. Sugitani, and M. Dijkhuizen. We develop general tools to show that a Noumi-Sugitani-Dijkhuizen type construction of quantum symmetric pairs can be performed preserving spherical representations from the classical situation. These tools apply to the symmetric pair FII and to all symmetric pairs which correspond to an automorphism of the underlying Dynkin diagram. Hence Noumi-Sugitani-Dijkhuizen type constructions with desirable properties are possible for various symmetric pairs for exceptional Lie algebras.

The internal dynamics of the exceptional Lie symmetry groups hierarchy and the coupling constants of unification

This short note focuses on the internal dynamics of exceptional Lie symmetry group hierarchies. Primitive invariants of harmonic two and three Stein compact and non-compact dual spaces are reconnected to the relevant Stein spaces of the first kind. We are able to deduce various fundamental inverse coupling constant for non-super symmetric and super symmetric unification of all fundamental interactions of high energy physics.

P-Adic analysis and the transfinite E8 exceptional Lie symmetry group unification

In P-Adic analysis like in a fractal Cantorian space there is no absolute scale. P-Adic analysis with its prime numbers base is the mathematical quarks of the exceptional E8 and E-infinity. The P-Adic space permits the use of Weyl original spacetime gauge theory which is the rationale behind E-infinity.

Exceptional Lie groups, E-infinity theory and Higgs Boson

In this paper we study the correlation between El-Naschie’s exceptional Lie groups hierarchies and his transfinite E-infinity space-time theory. Subsequently this correlation is used to calculate the number of elementary particles in the standard model, mass of the Higgs Bosons and some coupling constants.

Group gradings on ο(8, ℂ)

This is a matricial description of all the fourteen fine group gradings on the exceptional Lie algebra ο (8, ℂ).

Models of the Lie algebra [formula omitted

We describe the models of the exceptional Lie algebra F4 which are based on its semisimple subalgebras of rank 4. The underlying fact is that any reductive subalgebra of maximal rank of a simple Lie algebra induces a grading on this algebra by means of an abelian group, in such a way that the nontrivial components of the grading are irreducible modules.

Stein spaces in connection with El Naschie’s exceptional Lie groups hierarchies in high energy physics

In this work we introduce Kobayashi hyperbolic manifolds with their automorphism groups which are Lie groups. Bounded Stein manifolds are Kobayashi hyperbolic with Lie groups as automorphism groups. One and two-Stein spaces are in close connection with dimensions of El Naschie’s exceptional Lie groups hierarchies. The sum of seventeen one and two-Stein spaces can be expressed in terms of the dimensions of certain Lie groups. The number of high energy elementary particles in an extended standard model and their fuzzy values are found.