Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras

This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\subset W(S)$, fundamental polyhedron \Cal M of $W$ and an acceptable (corresponding to twisting coefficients) set P({\Cal M})\subset M of vectors orthogonal to faces of \Cal M (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra {\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M}))) which is graded by $M$. We show that \goth g has good behavior of imaginary roots, its denominator formula is defined in a natural domain and has good automorphic properties if and only if \goth g has so called {\it restricted arithmetic type}. We show that every finitely generated (i.e. P({\Cal M}) is finite) algebra {\goth g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1))) may be embedded to {\goth g}^{\prime\prime}(A(S,W,P({\Cal M}))) of the restricted arithmetic type. Thus, Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a natural class to study. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the best automorphic properties for the denominator function if they have {\it a lattice Weyl vector $\rho$}. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector $\rho$ are called {\it elliptic}Comment: Some corrections in Sects. 2.1, 2.2 were done. They don't reflect on results and ideas. 31 pages, no figures. AMSTe

Global anisotropy of arrival directions of ultra-high-energy cosmic rays: capabilities of space-based detectors

Planned space-based ultra-high-energy cosmic-ray detectors (TUS, JEM-EUSO and S-EUSO) are best suited for searches of global anisotropies in the distribution of arrival directions of cosmic-ray particles because they will be able to observe the full sky with a single instrument. We calculate quantitatively the strength of anisotropies associated with two models of the origin of the highest-energy particles: the extragalactic model (sources follow the distribution of galaxies in the Universe) and the superheavy dark-matter model (sources follow the distribution of dark matter in the Galactic halo). Based on the expected exposure of the experiments, we estimate the optimal strategy for efficient search of these effects.Comment: 19 pages, 7 figures, iopart style. v.2: discussion of the effect of the cosmic magnetic fields added; other minor changes. Simulated UHECR skymaps available at http://livni.inr.ac.ru/UHECRskymaps