Exceptional Simple Lie Groups Research Articles

Cooking pasta with Lie groups

We extend the (gauged) Skyrme model to the case in which the global isospin group (which usually is taken to be SU(N)) is a generic compact connected Lie group G. We analyze the corresponding field equations in (3+1) dimensions from a group theory point of view. Several solutions can be constructed analytically and are determined by the embeddings of three dimensional simple Lie groups into G, in a generic irreducible representation. These solutions represent the so-called nuclear pasta state configurations of nuclear matter at low energy. We employ the Dynkin explicit classification of all three dimensional Lie subgroups of exceptional Lie group to classify all such solutions in the case G is an exceptional simple Lie group, and give all ingredients to construct them explicitly. As an example, we construct the explicit solutions for G=G2. We then extend our ansatz to include the minimal coupling of the Skyrme field to a U(1) gauge field. We extend the definition of the topological charge to this case and then concentrate our attention to the electromagnetic case. After imposing a “free force condition” on the gauge field, the complete set of coupled field equations corresponding to the gauged Skyrme model minimally coupled to an Abelian gauge field is reduced to just one linear ODE keeping alive the topological charge. We discuss the cases in which such ODE belongs to the (Whittaker-)Hill and Mathieu types.

New non-naturally reductive Einstein metrics on exceptional simple Lie groups

In this article, we achieved several non-naturally reductive Einstein metrics on exceptional simple Lie groups, which are formed by the decomposition arising from general Wallach spaces. By using the decomposition corresponding to the two involutive automorphisms, we calculated the non-zero coefficients in the expression for the components of Ricci tensor with respect to the given metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gr\"{o}bner bases.

Cartan\u2019s incomplete classification and an explicit ambient metric of holonomy mathrm{G}_2^*

In his 1910 “Five Variables” paper, Cartan solved the equivalence problem for the geometry of (2, 3, 5) distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type mathrm{G}_2. He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least 6 (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution mathbf {E}. We produce a closed form for the Fefferman–Graham ambient metric {smash {{smash {widetilde{g}}}}}_{mathbf {E}} of the conformal class induced by (a real form of) mathbf {E}, expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of smash {{smash {widetilde{g}}}_{mathbf {E}}} is the exceptional group {smash {mathrm{G}}_2^*} and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general left-invariant conformal structures that were used in the determination of the explicit formula for smash {{smash {widetilde{g}}}_{mathbf {E}}}.

Non-naturally reductive Einstein metrics on exceptional Lie groups

Given an exceptional compact simple Lie group G we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of G over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Gröbner bases and classify the real solutions of the associated algebraic systems. For the Lie group G 2 we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups E 7 and E 8 , we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are Ad ( K ) -invariant by different Lie subgroups K .

Visible actions on flag varieties of exceptional groups and a generalization of the Cartan decomposition

We give a generalization of the Cartan decomposition for connected compact exceptional Lie groups motivated by the work on visible actions of T. Kobayashi [T. Kobayashi, J. Math. Soc. Japan 59 (2007) 669–691] for type A groups. This paper extends his results to the exceptional groups. First, we classify a pair of Levi subgroups (L,H) of any compact exceptional simple Lie group G such that G=LGσH where σ is a Chevalley–Weyl involution. This implies that the natural L-action on the generalized flag variety G/H is strongly visible, and likewise the H-action on G/L and the G-action on (G×G)/(L×H) are strongly visible. Second, we find a generalized Cartan decomposition G=LBH with B in Gσ by using the herringbone stitch method which was introduced by Kobayashi. Applications to multiplicity-free representations are also discussed.

Formula omitted]-root systems of exceptional flag manifolds

T-roots of a flag manifold M=G/H of a compact semisimple Lie group G are the characters of the torus Z(H) corresponding to the one-dimensional Z(H)-invariant subspaces of C⊗(g/h).We classify the systems of T-roots of the flag manifolds M of the exceptional compact simple Lie groups with the second Betti number b2(M)⩾2.

Exceptional and semi simple Lie groups hierarchies and the maximum number of elementary particles beyond the standard model of high energy physics

In this work and following some major developments in our understanding of the connection between the exceptional Lie symmetry groups hierarchy and physics, we present finite and infinite dimensional Lie groups and their connections to the orthogonal and unitary groups. The dimension of the Lie group E 7 1 2 is included in the derivation of the number of elementary particle-like super states 576 and 672. Connections to the dimensions of Lie groups and the 72 and 84 elementary particles recently considered by the Author are discussed.

MULTIPLE HAMILTONIAN STRUCTURE OF BOGOYAVLENSKY–TODA LATTICES

This paper is mainly a review of the multi-Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi-Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.

Realizations of subgroups of type $D_{8}$ of connected exceptional simple Lie groups of type $E_{8}$

In [4],we realized subgroups of type A%, A4 x A4 and A2 x E$ of the compact simple Lie group of type E%. In thispaper, we shallrealize subgroups of type D% of the compact and non-compact simple Lie groups of type E%. In [5], [6], [10] and [11], Yokota and some members of his school found all involutive automorphisms a and realized subgroups Ga of fixed points of connected exceptional simple Lie groups G explicitly, which correspond to Berger's result of simple Lie algebras [2]. But in their results concerning subgroups of type Dg of Lie groups of type E%, the definition of subgroups are not clear and proof of isomorphism is very difficultin comparison with their other results. We improve those resultsin thispaper. Our improvement make results that are more simple and intelligible.Hence they are of widely applicable to symmetric spaces. Our resultsare as follows.

The place of the adjoint representation in the Kronecker square of irreducible representations of simple Lie groups

The multiplicity of occurrence of the adjoint representation in the decomposition of the square of any finite-dimensional irreducible representation of any compact simple Lie group is shown to be equal to the number of non-vanishing components of the Dynkin label of . The resolution of this multiplicity into contributions to the symmetric and antisymmetric squares of is discussed, with complete results being found for all of the classical and some of the exceptional simple Lie groups, and partial results culminating in conjectures for the remaining exceptional groups.

The embeddings of discrete series into principal series for an exceptional real simple Lie group of type $G_2$

The embeddings of discrete series into principal series for an exceptional real simple Lie group of type $G_2$

The embeddings of discrete series into principal series for an exceptional real simple Lie group of type $G_2$

The embeddings of discrete series into principal series for an exceptional real simple Lie group of type $G_2$

Internal, External, and Generalized Symmetries

Bácklund′s theorem, which characterizes contact transformations, is generalized to give an analogous characterization of "internal symmetries" of systems of differential equations. For a wide class of systems of differential equations, we prove that every internal symmetry comes from a first order generalized symmetry and, conversely, every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry. We analyze the contact conditions in detail, deducing powerful necessary conditions for a system of differential equations that admit "genuine" internal symmetries, i.e., ones which do not come from classical "external" symmetries. Applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to Hilbert and Cartan are the noncompact real form of the exceptional simple Lie group G 2.

Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams

The Voronoi domains, their duals (Delaunay domains) and all their faces of any dimension are classified and described in terms of the Weyl group action on a representative of each type of face. The representative of a face type is specified by a decoration of the corresponding Coxeter-Dynkin diagram. The rules of domain description are uniform for root lattices of simple Lie groups of all types. An explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple Lie groups. The Coxeter-Dynkin diagrams required here are the diagrams extended by the highest short root. Each diagram is partitioned into two subdiagrams, one describing completely a d-face of the Voronoi domain, its complement completely describing the dual of the d-face. The applicability of the authors' classification method to generalized kaleidoscopes is explained.

Nonlinear equations with superposition formulas and the exceptional group G2. I. Complex and real forms of g2 and their maximal subalgebras

In order to study nonlinear ordinary differential equations with superposition principles, related to the exceptional simple Lie group G2, the complex and real forms of its Lie algebra are examined and their maximal subalgebras are summarized. In particular the parabolic subalgebras of the noncompact real form gNC2(R) are determined. Explicit matrix realizations of the fundamental representation D(1,0) are used and studied in connection with invariant subspaces in a seven-dimensional (complex or real) vector space. The results are collected in three tables of specific interest for the study of nonlinear differential equations, which will be developed in Paper II of this series.

Alternating 3-Forms and Exceptional Simple Lie Groups of Type G2

It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G2 as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let W be a complex 7-dimensional vector space. Among the alternating 3-forms on W there is a connected dense open subset Ψ(W) of “maximal” forms. If ψ ∈ Ψ(W) then the subgroup of AUTC(W) consisting of the invertible complex-linear transformations S such that ψ(S•, S•, S•) = ψ(•, •, •) is denoted G(ψ), and, in Proposition 3.6. we provewhere G1(ψ) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra of type G2 is defined in terms of the alternating 3-form ψ alone without the need to specify an invariant quadratic form. In the real case the result is more striking.

The Weyl groups and weight multiplicities of the exceptional Lie groups

The Weyl group, WG, of each exceptional simple Lie group G, is described in detail. Its structure is defined in terms of its coset decomposition with respect to the Weyl group, WH, of a classical semi-simple Lie group, H, embedded naturally in G. The concepts of G-dominance and G-equivalence are defined and used to determine, from the character formula of Weyl, the branching rule associated with the restriction of group elements from G to H. The Weyl group WG is used further to impose constraints on both the branching multiplicities for G to H and the weight multiplicities of G. These constraints are used to evaluate the weight multiplicities of F4, E6, E7 and E8 together with the branching multiplicities for E8 to SO(16).