Infinite Dimensional Chevalley Groups and Kac-moody Groups over Z
نویسنده
چکیده
Let A be a symmetrizable generalized Cartan matrix. Let g be the corresponding Kac-Moody algebra. Let G(R) be Tits’ Kac-Moody group functor over commutative rings R associated to g. Then G(R) is given by five axioms KMG1-KMG5 which are natural extensions of the properties of Chevalley-Demazure group schemes. We give a construction of the Tits functor for symmetrizable Kac-Moody groups G using integrable highest weight modules for the corresponding Kac-Moody algebra and a Z-form of the universal enveloping algebra. This gives a construction of Kac-Moody groups G as infinite dimensional analogs of finite dimensional Chevalley groups. The integral form G(Z) is the value of the Tits functor over Z. We give an explicit construction of the Tits functor G(Z) using integrable highest weight modules for g and a Z-form of the universal enveloping algebra of g and we show that this construction satisfies the axioms KMG1-KMG5 of Tits. We prove a Bruhat decomposition G(Q) = G(Z)B(Q) over Q and we give several generating sets for G(Z), including two finite minimal generating sets.
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