Modular realizations of hyperbolic Weyl groups. (English) Zbl 1280.81064
The paper is devoted to the study of recently discovered generic isomorphisms between the Weyl groups of hyperbolic Kac-Moody algebras and modular groups over integer domains in normed division algebras. As the authors declare, this over fifty pages long work is rather a first step towards the comprehensive and complete theory. It contains numerous new results. The topic is of great interest, also due to the potential applications in quantum gravity and M-theory. The authors find realizations of the group action on generalized upper half-planes over the division algebras and construct explicitly automorphic forms for quaternionic and octonionic cases. Such generalized upper half-planes are associated with the modular groups appearing in the isomorphisms. Among the main new results is the explicit octavian realization of the even Weyl group \(W^+(E_{10})\) and then of \(W^+(E_{8})\) as its special case.
The paper consists of six sections and four appendices. From the contents: Introduction; Weyl groups as modular groups; \(D_4\) and its hyperbolic extension; \(E_8\) and its hyperbolic extension; Automorphic functions on \(\mathcal{H}(\mathbb{A})\).
The paper consists of six sections and four appendices. From the contents: Introduction; Weyl groups as modular groups; \(D_4\) and its hyperbolic extension; \(E_8\) and its hyperbolic extension; Automorphic functions on \(\mathcal{H}(\mathbb{A})\).
Reviewer: Andrzej Frydryszak (Wrocław)
MSC:
17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
83C45 | Quantization of the gravitational field |
11F22 | Relationship to Lie algebras and finite simple groups |