A study of infinite dimensional algebraic groups and Lie algebras, and an application to words and sequences

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In the case when rank 2 hyperbolic Kac-Moody groups have trivial commutation relations, we obtained that the groups over certain infinite fields are simple modulo their centers. That is, if we suppose that a rank 2 hyperbolic Cartan matrix has no -1 entry, and if we choose the algebraic closure of a finite field, then we can obtain the simplicity of our group. We also classified the locally affine Lie algebras. Furthermore, we showed that there is an interesting correspondence between fundamental automata and hierarchy of numbers.