Modular realizations of hyperbolic Weyl groups

We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group

Modular realizations of hyperbolic Weyl groups

We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group

Labor Movements, Mainly in Eastern Europe

The main factors of labor movements, mainly in Eastern Europe, are considered in this paper. The influence of economic indicators and peculiarities of economic development of the countries of the region in the current migration situation in Europe as a whole has been studied. The analysis of the migration policy of the countries of the European Union and individual countries is performed. The main tendencies of migration flows of the population in the region and separate countries of Eastern Europe are revealed. General migration problems as an integrational form of the existence of modern economies of the countries are summarized

Remarks on E11 approach

We consider a few topics in $E_{11}$ approach to superstring/M-theory: even subgroups ($Z_2$ orbifolds) of $E_{n}$, n=11,10,9 and their connection to Kac-Moody algebras; $EE_{11}$ subgroup of $E_{11}$ and coincidence of one of its weights with the $l_1$ weight of $E_{11}$, known to contain brane charges; possible form of supersymmetry relation in $E_{11}$; decomposition of $l_1$ w.r.t. the $SO(10,10)$ and its square root at first few levels; particle orbit of $l_1 \ltimes E_{11}$. Possible relevance of coadjoint orbits method is noticed, based on a self-duality form of equations of motion in $E_{11}$.Comment: Two references adde

Sugawara-type constraints in hyperbolic coset models

In the conjectured correspondence between supergravity and geodesic models on infinite-dimensional hyperbolic coset spaces, and E10/K(E10) in particular, the constraints play a central role. We present a Sugawara-type construction in terms of the E10 Noether charges that extends these constraints infinitely into the hyperbolic algebra, in contrast to the truncated expressions obtained in arXiv:0709.2691 that involved only finitely many generators. Our extended constraints are associated to an infinite set of roots which are all imaginary, and in fact fill the closed past light-cone of the Lorentzian root lattice. The construction makes crucial use of the E10 Weyl group and of the fact that the E10 model contains both D=11 supergravity and D=10 IIB supergravity. Our extended constraints appear to unite in a remarkable manner the different canonical constraints of these two theories. This construction may also shed new light on the issue of `open constraint algebras' in traditional canonical approaches to gravity.Comment: 49 page

Generalised holonomies and K(E9)

The involutory subalgebra K(E$_9$) of the affine Kac-Moody algebra E$_9$ was recently shown to admit an infinite sequence of unfaithful representations of ever increasing dimensions arXiv:2102.00870. We revisit these representations and describe their associated ideals in more detail, with particular emphasis on two chiral versions that can be constructed for each such representation. For every such unfaithful representation we show that the action of K(E$_9$) decomposes into a direct sum of two mutually commuting (`chiral' and `anti-chiral') parabolic algebras with Levi subalgebra $\mathfrak{so}(16)_+\,\oplus\,\mathfrak{so}(16)_-$. We also spell out the consistency conditions for uplifting such representations to unfaithful representations of K(E$_{10}$). From these results it is evident that the holonomy groups so far discussed in the literature are mere shadows (in a Platonic sense) of a much larger structure

E10 and SO(9,9) invariant supergravity

We show that (massive) D=10 type IIA supergravity possesses a hidden rigid SO(9,9) symmetry and a hidden local SO(9) x SO(9) symmetry upon dimensional reduction to one (time-like) dimension. We explicitly construct the associated locally supersymmetric Lagrangian in one dimension, and show that its bosonic sector, including the mass term, can be equivalently described by a truncation of an E10/K(E10) non-linear sigma-model to the level \ell<=2 sector in a decomposition of E10 under its so(9,9) subalgebra. This decomposition is presented up to level 10, and the even and odd level sectors are identified tentatively with the Neveu--Schwarz and Ramond sectors, respectively. Further truncation to the level \ell=0 sector yields a model related to the reduction of D=10 type I supergravity. The hyperbolic Kac--Moody algebra DE10, associated to the latter, is shown to be a proper subalgebra of E10, in accord with the embedding of type I into type IIA supergravity. The corresponding decomposition of DE10 under so(9,9) is presented up to level 5.Comment: 1+39 pages LaTeX2e, 2 figures, 2 tables, extended tables obtainable by downloading sourc

Representations of G+++ and the role of space-time

We consider the decomposition of the adjoint and fundamental representations of very extended Kac-Moody algebras G+++ with respect to their regular A type subalgebra which, in the corresponding non-linear realisation, is associated with gravity. We find that for many very extended algebras almost all the A type representations that occur in the decomposition of the fundamental representations also occur in the adjoint representation of G+++. In particular, for E8+++, this applies to all its fundamental representations. However, there are some important examples, such as An+++, where this is not true and indeed the adjoint representation contains no generator that can be identified with a space-time translation. We comment on the significance of these results for how space-time can occur in the non-linear realisation based on G+++. Finally we show that there is a correspondence between the A representations that occur in the fundamental representation associated with the very extended node and the adjoint representation of G+++ which is consistent with the interpretation of the former as charges associated with brane solutions.Comment: 45 pages, 9 figures, 9 tables, te

COMMERCIAL FISHING RIGHTS ALLOCATION IN POST-APARTHEID SOUTH AFRICA: RECONCILING EQUITY AND STABILITY

No Abstract.African Journal of Marine Science Vol.25(1) 2003: pp. 25-3