1994/12/06 by Viacheslav V. Nikulin, Nikulin, Viacheslav V. · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.alg-geom/9412003
openalex publication_date 1994/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We name an indecomposable symmetrizable generalized Cartan matrix A and the corresponding Kac--Moody Lie algebra \goth g ^′ (A) \it of the arithmetic type if for any β∈ Q with (β| β)<0 there exist n(β)∈ \Bbb N and an imaginary root α∈ Δim such that n(β)β≡ α\mod Ker (.|.) on Q. Here Q is the root lattice. This generalizes "symmetrizable hyperbolic" type of Kac and Moody. We show that generalized Cartan matrices of the arithmetic type are divided in 4 types: (a) finite, (b) affine, (c) rank two, and (d) arithmetic hyperbolic type. The last type is very closely related with arithmetic groups generated by reflections in hyperbolic spaces with the field of definition \Bbb Q. We apply results of the author and É.B. Vinberg on arithmetic groups generated by reflections in hyperbolic spaces to describe generalized Cartan matrices of the arithmetic hyperbolic type and to show that there exists a finite set of series of the generalized Cartan matrices of the arithmetic hyperbolic type. For the symmetric case all these series are known.