双曲空间中的反射群与Lorentzian Kac-Moody李代数的分母公式
Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras
摘要：
这是我们的“讲座卡茨-穆迪李代数的算术型”\cite{25}的延续。我们考虑双曲（即签名$(n,1)$）积分对称双线性形式$S:M\times M \to {\Bbb Z}$（即双曲格），反射群$W\subset W(S)$，基本多面体$\Cal M$的$W$和一个可接受的（对应于扭曲系数）集$P({\Cal M})\subset M$的矢量正交的面$\Cal M$（简单的根）。可以构造相应的Lorentzian Kac-Moody李代数${\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$，它是由$M$分次的.证明了$\goth g$具有良好的虚根性质，其分母公式定义在自然区域上，且具有良好的自守性质当且仅当$\goth g$具有所谓的{ \it限制算术型}.我们证明了每一个可生成的（即$P({\Cal M})$是有限的）代数${\goth g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1)))$都可以嵌入到限制算术类型的${\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$中。因此，Lorentzian Kac-Moody李代数的限制算术型是一个自然的类研究。限制算术型的Lorentzian Kac-Moody李代数具有分母函数的最佳自守性质，如果它们具有{ \it a lattice Weyl vector $ρ$ }。具有广义格Weyl向量$ρ$的限制算术型Lorentzian Kac-Moody李代数称为{ \it椭圆}
英文摘要：
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 $ρ$}. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector $ρ$ are called {\it elliptic}
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