Decompositions of hyperbolic Kac-Moody algebras with respect to imaginary root groups
arxiv(2024)
摘要
We propose a novel way to define imaginary root subgroups associated with
(timelike) imaginary roots of hyperbolic Kac-Moody algebras. Using in an
essential way the theory of unitary irreducible representation of covers of the
group SO(2,1), these imaginary root subgroups act on the complex Kac-Moody
algebra viewed as a Hilbert space. We illustrate our new view on Kac-Moody
groups by considering the example of a rank-two hyperbolic algebra that is
related to the Fibonacci numbers. We also point out some open issues and new
avenues for further research, and briefly discuss the potential relevance of
the present results for physics and current attempts at unification.
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