On Classification of Lorentzian Kac–moody Algebras
نویسندگان
چکیده
We discuss a general theory of Lorentzian Kac–Moody algebras which should be a hyperbolic analogy of the classical theories of finite-dimensional semisimple and affine Kac–Moody algebras. First examples of Lorentzian Kac–Moody algebras were found by Borcherds. We consider general finiteness results about the set of Lorentzian Kac–Moody algebras and the problem of their classification. As an example, we give classification of Lorentzian Kac–Moody algebras of the rank three with the hyperbolic root lattice S t , symmetry lattice L ∗ t , and the symmetry group Ô(Lt), t ∈ N, where
منابع مشابه
A Theory of Lorentzian Kac–moody Algebras
We present a variant of the Theory of Lorentzian (i. e. with a hyperbolic generalized Cartan matrix) Kac–Moody algebras recently developed by V. A. Gritsenko and the author. It is closely related with and strongly uses results of R. Borcherds. This theory should generalize well-known Theories of finite Kac–Moody algebras (i. e. classical semi-simple Lie algebras corresponding to positive genera...
متن کاملAlmost split real forms for hyperbolic Kac-Moody Lie algebras
A Borel-Tits theory was developped for almost split forms of symmetrizable Kac-Moody Lie algebras [J. of Algebra 171, 43-96 (1995)]. In this paper, we look to almost split real forms for symmetrizable hyperbolic KacMoody Lie algebras and we establish a complete list of these forms, in terms of their Satake-Tits index, for the strictly hyperbolic ones and for those which are obtained as (hyperbo...
متن کاملReflection 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” [25]. We consider hyperbolic (i.e. signature (n, 1)) integral symmetric bilinear form S : M × M → Z (i.e. hyperbolic lattice), reflection group W ⊂ W (S), fundamental polyhedron M of W and an acceptable (corresponding to twisting coefficients) set P (M) ⊂ M of vectors orthogonal to faces of M (simple roots)...
متن کاملSpacelike Singularities and Hidden Symmetries of Gravity
We review the intimate connection between (super-)gravity close to a spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, whi...
متن کاملCosmological Singularities, Billiards and Lorentzian Kac-Moody Algebras
The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of BelinskiiKhalatnikov-Lifshitz and technically simplified by the use of the ArnowittDeser-Misner Hamiltonian formalism) that the asymptotic behaviour, as one approaches the singularity, of the general solutio...
متن کامل