62 research outputs found
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 ) integral symmetric bilinear
form (i.e. hyperbolic lattice), reflection group
, fundamental polyhedron \Cal M of 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 .
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 }. Lorentzian Kac--Moody Lie algebras of the
restricted arithmetic type with generalized lattice Weyl vector 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
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