2,631 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
The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
We find automorphic corrections for the Lorentzian Kac--Moody algebras with
the simplest generalized Cartan matrices of rank 3:
A_{1,0} =
2 0 -1
0 2 -2
-1 -2 2
and
A_{1,I} =
2 -2 -1
-2 2 -1
-1 -1 2
For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form
\chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30
with respect to a 2-congruence subgroup. We find infinite product or sum
expansions for these forms. Our method of construction of \chi_{35} leads to
the direct construction of Siegel modular forms by infinite product expansions,
whose divisors are the Humbert surfaces with fixed discriminants. Existence of
these forms was proved by van der Geer in 1982 using some geometrical
consideration. We announce a list of all hyperbolic symmetric generalized
Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a
lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.Comment: 40 pages, no figures. AMS-Te
Kahlerian K3 surfaces and Niemeier lattices
Using results (especially see Remark 1.14.7) of our paper "Integral symmetric
bilinear forms and some of their applications", 1979, we clarify relation
between Kahlerian K3 surfaces and Niemeier lattices. We want to emphasise that
all twenty four Niemeier lattices are important for K3 surfaces, not only the
one which is related to the Mathieu group.Comment: Var7: 88 pages. We added last case
Neurophysiophenomenology – predicting emotional arousal from brain arousal in a virtual reality roller coaster
Arousal is a core affect constituted of both bodily and subjective states that prepares an agent to respond to events of the natural environment. While the peripheral physiological components of arousal have been examined also under naturalistic conditions, its neural correlates were suggested mainly on the basis of simplifed experimental designs. We used virtual reality (VR) to present a highly immersive and contextually rich scenario of roller coaster rides to evoke naturalistic states of emotional arousal. Simultaneously, we recorded EEG to validate the suggested neural correlates of arousal in alpha frequency oscillations (8-12Hz) over temporo-parietal cortical areas. To fnd the complex link between these alpha components and the participants’ continuous subjective reports of arousal, we employed a set of complementary analytical methods coming from machine learning and deep learning
A lecture on Arithmetic Mirror Symmetry and Calabi-Yau manifolds
We extend our variant of mirror symmetry for K3 surfaces \cite{GN3} and
clarify its relation with mirror symmetry for Calabi-Yau manifolds. We
introduce two classes (for the models A and B) of Calabi-Yau manifolds fibrated
by K3 surfaces with some special Picard lattices. These two classes are related
with automorphic forms on IV type domains which we studied in our papers
\cite{GN1}-\cite{GN6}. Conjecturally these automorphic forms take part in the
quantum intersection pairing for model A, Yukawa coupling for model B and
mirror symmetry between these two classes of Calabi-Yau manifolds. Recently
there were several papers by physicists where it was shown on some examples. We
propose a problem of classification of introduced Calabi-Yau manifolds. Our
papers \cite{GN1}-\cite{GN6} and \cite{N3}-\cite{N14} give a hope that this is
possible. They describe possible Picard or transcendental lattices of general
K3 fibers of the Calabi-Yau manifolds.Comment: AMS-Tex, 11 pages, no figures. The variant prepared for publication;
many small changes introduce
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