Lie Group F4 Research Articles

Accelerated discovery of machine-learned symmetries: Deriving the exceptional Lie groups G2, F4 and E6

Recent work has applied supervised deep learning to derive continuous symmetry transformations that preserve the data labels and to obtain the corresponding algebras of symmetry generators. This letter introduces two improved algorithms that significantly speed up the discovery of these symmetry transformations. The new methods are demonstrated by deriving the complete set of generators for the unitary groups U(n) and the exceptional Lie groups G2, F4, and E6. A third post-processing algorithm renders the found generators in sparse form. We benchmark the performance improvement of the new algorithms relative to the standard approach. Given the significant complexity of the exceptional Lie groups, our results demonstrate that this machine-learning method for discovering symmetries is completely general and can be applied to a wide variety of labeled datasets.

Trace dynamics and division algebras: towards quantum gravity and unification

Abstract We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang–Mills fields, and fermions. Dynamical variables are described by odd-grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper, we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G 2, replace space-time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G 2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1) em , describes the electrocolour symmetry, as shown earlier by Furey. We predict a new massless spin one boson (the ‘Lorentz’ boson) which should be looked for in experiments. Our Lagrangian correctly describes three fermion generations, through three copies of the group G 2, embedded in the exceptional Lie group F 4. This is the unification group for the four fundamental interactions, and it also happens to be the automorphism group of the exceptional Jordan algebra. Gravitation is shown to be an emergent classical phenomenon. Although at the Planck scale, there is present a quantised version of the Lorentz symmetry, mediated by the Lorentz boson, we argue that at sub-Planck scales, the self-adjoint part of the octonionic trace dynamics bears a relationship with string theory in 11 dimensions.

Einstein–Randers metrics on some homogeneous manifolds

In this paper we get some new examples of non-Riemannian Einstein–Randers metrics on homogeneous manifolds G/H when G is either the compact simple Lie group F4 or G2. Based on the discussions associated with Kähler C-spaces, we obtain some new Einstein Riemannian metrics on some homogeneous manifolds and prove that they admit non-Riemannian Einstein–Randers metrics.

The Image of the Lepowsky Homomorphism for the Group F

Let Go be a semisimple Lie group, let Ko be a maximal compact subgroup of Go and let denote the complexification of their Lie algebras. Let G be the adjoint group of and let K be the connected Lie subgroup of G with Lie algebra ⁠. If is the universal enveloping algebra of ⁠, then will denote the centralizer of K in ⁠. Also let be the projection map corresponding to the direct sum associated to an Iwasawa decomposition of Go adapted to Ko. In this paper, we give a characterization of the image of under the injective antihomorphism ⁠, considered by Lepowsky in [12], when Go is isomorphic to the rank 1 real form F of the exceptional Lie group F4.

ON THE GEOMETRY OF THE SUPERMULTIPLET IN M-THEORY

The massless supermultiplet of 11-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F4 into those of its maximal compact subgroup Spin(9). In an earlier paper, a dynamical Kaluza–Klein origin of this observation is proposed with internal space the Cayley plane, 𝕆P2, and topological aspects are explored. In this paper we consider the geometric aspects and characterize the corresponding forms which contribute to the action as well as cohomology classes, including torsion, which contribute to the partition function. This involves constructions with bilinear forms. The compatibility with various string theories are discussed, including reduction to loop bundles in ten dimensions.

${\mathbb O}P^2$ bundles in M-theory

Ramond has observed that the massless multiplet of eleven-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F4 into those of its maximal compact subgroup Spin(9). The possibility of a topological origin for this observation is investigated by studying Cayley plane, OP2, bundles over eleven-manifolds Y. The lift of the topological terms gives constraints on the cohomology of Y which are derived. Topological structures and genera on Y are related to the corresponding ones on the total space M. The latter, being 27-dimensional, might provide a candidate for `bosonic M-theory'. The discussion leads to a connection with an octonionic version of Kreck-Stolz elliptic homology theory.

Mapping the geometry of the F4 group

In this paper, we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 Hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2 = F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.

Tubular Structures of Compact Symmetric Spaces Associated with the Exceptional Lie Group F4

In the present paper we discuss in detail cohomogeneity one isometric actions on the compact symmetric spaces F4 and \(F_{4}\backslash({\rm Sp}(3)\,\times\, {\rm Sp}(1)\backslash {\mathbb Z} _2)\) . By means of isometric actions we show that these symmetric spaces have got well-defined tubular structures around the totally geodesic submanifolds Spin(9) and \(G^+_4({\mathbb R}^9)\) , respectively. Therefore these symmetric spaces can be thought of as compact tubes. The radii of the tubes and the principal curvatures of the tubular hypersurfaces are determined in explicit form. Moreover, we apply these results to compute the volumes of the principal orbits and the volumes of the symmetric spaces associated with the compact Lie group F4.

On the homotopy type of the classifying space of the exceptional Lie group F4

Previous work of several authors shows that the exceptional Lie group of rank 4, F4, as a p-compact group, is determined up to isomorphism by the isomorphism type of its maximal torus normalizer for p > 2. This paper considers the case p= 2 proving that F4 as 2-compact group is also determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows the authors to determine the integral homotopy type of F4 among connected finite loop spaces with maximal tori.

Reflection Groups on the Octave Hyperbolic Plane

For two different integral formsKof the exceptional Jordan algebra we show that AutKis generated by octave reflections. These provide “geometric” examples of discrete reflection groups acting with finite covolume on the octave (or Cayley) hyperbolic plane OH2, the exceptional rank one symmetric space. (The isometry group of the plane is the exceptional Lie groupF4(−20).) Our groups are defined in terms of Coxeter's discrete subring K of the nonassociative division algebra O and we interpret them as the symmetry groups of “Lorentzian lattices” over K. We also show that the reflection group of the “hyperbolic cell” over K is the rotation subgroup of a particularrealreflection group acting onH8≅OH1. Part of our approach is the treatment of the Jordan algebra of matrices that are Hermitian with respect to any real symmetric matrix.

Diagonalization of an element P of $\mathfrak{P}^{c}$ by the compact Lie group $E_{7}$

The following diagonalization problems are well-known. A Hermitian matrix X e M(n,K) over K = R,C,H can be transformed to a diagonal form by some element A of the groups SO(n), SU(n), Sp(n), respectively.Moreover we know that a Hermitian matrix X e M(3,K) over K = (£,(Ec can be transformed to a diagonal form by some element a of the compact Lie groups F4, E&, respectively (Freudenthal [1],Yokota [9]).Now, in this paper, we show that an element P of ^5C (which is a minimal dimensional representation space of the compact Lie group E-j) can be transformed to a diagonal form by some element a of £7 (Theorem 5).In the last section, we give a canonical diagonal form of P of ^pc (Theorem 12). We would like to thank T. Miyashita for his calculation of Lemma 3.

Geometry of Planes over Nonassociative Algebras

In this paper the geometric interpretation of the exceptional Lie groups F4, E6, E7, and E8 is given. These groups are groups of motions of elliptic hyperbolic planes over nonassociative algebras of octaves and split octaves and their tensor products with algebras of usual and split complex numbers, quaternions and octaves. The explicit expressions of motions of these planes and their figures of symmetry are presented.

Mathematics of f electrons

Unexpected simplifications in the theory of the atomic f shell are commonplace. They usually take the form of matrix elements of physical operators that exhibit relationships that go beyond what would be expected on the basis of the Wigner–Eckart theorem. The general problem is discussed from several viewpoints. Kramers' spin invariants and the use of octonions lead to various interesting connections, but do not seem to bring a solution to the central problem any closer. The quark model is applied to a study of two sets of fractional parentage coefficients. Applications of the Lie group F4 to the analysis of part of the spin-other-orbit interaction suggest other areas for future work. The usefulness of drawing correspondences between antiHermitian operators that possess the same group labels as Hermitian operators of physical interest is explored.

Finite Subgroups of F 4 (C) and E 6 (C)

The isomorphism types of finite Lie primitive subgroups of the complex Lie groups E6(C) and F4(C) are determined. Here, we call a finite subgroup of a complex Lie group G Lie primitive if it is not contained in a proper closed subgroup of G of positive dimension. Induction can be used to investigate subgroups which are not Lie primitive. Some additional information is provided, such as the characters of these finite subgroups on some small-dimensional modules for the Lie groups. In studying these groups, we mainly use two rational linear representations of the universal covering group E ~ of E6(C), namely a 27-dimensional module (there are two inequivalent ones), denoted by K, and the adjoint module. In particular, we make heavy use of the characters of E ~ on these modules. The group F4(C) occurs in E ~ as the stabilizer subgroup of a vector in K. The finite simple groups of which a perfect central extension occurs in F4(C) or E6(C) are: via G2: Alt5, Alt6, L(2,7), L(2,8), L(2,13), U(3,3), via F4: Alt7, Alt8, Alt9, L(2,17), L(2,25), L(2,27), L(3,3), 3 D 4 ( 2 ), U(4,2), O(7,2), O+(8,2), via E6: Alt10, Alt11, L(2, 11), L(2,19), L(3,4), U(4,3), 2 F 4 ( 2 ) ′ , M11, J2. This list has been found using the classification of the finite simple groups. On the basis of this list, the finite Lie primitive subgroups are found to be either the normalizers of one of these subgroups or of one of the two elementary abelian 3-groups found by Alekseevskii. 1991 Mathematics Subject Classification: 20K47, 20G40, 17B45, 20C10.

3-primary 𝑣₁-periodic homotopy groups of 𝐹₄ and 𝐸₆

We compute the 3-primary v 1 {v_1} -periodic homotopy groups of the exceptional Lie groups F 4 {F_4} and E 6 {E_6} . The unstable Novikov spectral sequence is used for the most delicate part of the analysis.

Three contributions to the homotopy theory of the exceptional Lie groups G2 and F4

Three contributions to the homotopy theory of the exceptional Lie groups G2 and F4

Satake diagrams, Iwasawa decompositions, and representations of the exceptional Lie group F4(−20)

The parabolic subgroups of the noncompact, exceptional Lie group F4(−20) are computed from a systematic analysis of Iwasawa and Langlands decompositions. Satake diagrams have been used to determine the involutive automorphisms of F4(−20) which facilitate the Iwasawa decompositions. The polarizations associated with noncompact orbits with parabolic subgroups are computed. The representations so obtained for these polarizations using Kostant’s induction scheme yield the principal series representations of F4(−20).