A Matsumoto-type theorem for Kac-Moody groups
A presentation of the \(K_ 2\) group associated with a Kac-Moody group over a field is given, which is a natural generalization of the result by \textit{H. Matsumoto} [Ann. Sci. Ecole Norm. Sup., IV. Ser. 2, 1-62 (1969; Zbl 0261.20025)] for a split semisimple algebraic group. Also a universal central extension of a Kac-Moody group over an infinite field is determined using the same method as \textit{R. Steinberg} [Lectures on Chevalley groups, Yale Univ. Lect. Note, New Haven, Conn. (1968)]. Hence the Schur multiplier of a (universal) Kac-Moody group over an infinite field is just the corresponding \(K_ 2\) group. The main result is as follows. Let \(A=(a_{ij})\) be an \(n\times n\) generalized Cartan matrix, F a field, and \(K_ 2(A,F)\) the \(K_ 2\) group associated with a Kac- Moody group over F defined by A. Then \(K_ 2(A,F)\) is isomorphic to the abelian group generated by the symbols \(c_ i(u,v)\) for \(1\leq i\leq n\) and u, v in the multiplicative group \(F^{\times}\) of F with the following defining relations: \(c_ i(u,v)c_ i(u,vw)=c_ i(uv,w)c_ i(v,w)\), \(c_ i(1,1)=1\), \(c_ i(u,v)=c_ i(u^{-1},v^{-1})\), \(c_ i(u,v)=c_ i(u,(1-u)v)\) if \(u\neq 1\), \(c_ i(u,v^{a_{ji}})=c_ j(u^{a_{ij}},v)\), \(c_{ij}(uv,w)=c_{ij}(u,w)c_{ij}(v,w)\) for all \(1\leq i\neq j\leq n\) and \(u,v,w\in F^{\times}\), where \(c_{ij}(u,vw)\) is defined to be \(c_ i(u,v)c_ j(u,w)\).
presentation of the \(K_ 2\) group associated with a Kac-Moody group over a field
universal central extension of a Kac-Moody group over an infinite field
scientific article; zbMATH DE number 4148385
scientific article