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

On the Topological Classification of Real Enriques Surfaces. I

This note contains preliminary calculation of topological types or real Enriques surfaces. We realize 59 topological types of real Enriques surfaces (Theorem 6) and show that all other topological types belong to the list of 21 topological types (Theorem 7). In fact, our calculation contains much more information which is probably useful to constract or prohibit unknown topological types.Comment: 15 pages, Ams-Tex Version 2.

On the classification of hyperbolic root systems of the rank three. Part I

It was recently understood that from the point of view of automorphic Lorentzian Kac-Moody algebras and some aspects of Mirror Symmetry, interesting hyperbolic root systems should have restricted arithmetic type and a generalized lattice Weyl vector. One can consider hyperbolic root systems with these properties as an appropriate hyperbolic analog of the classical finite and affine root systems. This series of papers is devoted to classification of hyperbolic root systems of restricted arithmetic type and with a generalized lattice Weyl vector $\rho$, having the rank 3 (it is the first non-trivial rank). In the Part I we announce classification of the maximal hyperbolic root systems of elliptic (i.e. $\rho^2 >0$) and parabolic (i.e. $\rho^2=0$) type, having the rank 3. We give sketch of the proof. Details of the proof and further results (non-maximal cases) will be given in Part II. Classification for hyperbolic type (i.e. $\rho^2<0$) and applications will be considered in Part III.Comment: AMS-Tex, 58 pages, no figure