Uniqueness of representation--theoretic hyperbolic Kac--Moody groups over \Z

2015/12/15 by Lisa Carbone, Frank Wagner, Carbone, Lisa +1
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT

paper · pdf · doi:10.48550/arxiv.1512.04623

arxiv created 2016/02/06 · arxiv updated 2016/02/09

Abstract

For a simply laced and hyperbolic Kac--Moody group G=G(R) over a commutative ring R with 1, we consider a map from a finite presentation of G(R) obtained by Allcock and Carbone to a representation--theoretic construction G^λ(R) corresponding to an integrable representation V^λ with dominant integral weight λ. When R=\Z, we prove that this map extends to a group homomorphism ρ_λ,\Z: G(\Z) → G^λ(\Z). We prove that the kernel K^λ of the map ρ_\lam,\Z: G(\Z)→ G^\lam(\Z) lies in H(\C) and if the group homomorphism φ:G(\Z)→ G(\C) is injective, then K^λ≤ H(\Z)≅(\Z/2\Z)^rank(G).

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