A Matsumoto-type theorem for Kac-Moody groups
A Matsumoto-type theorem for Kac-Moody groups
复制标题
Kac-Moody 群的 Matsumoto 型定理
DOI：
10.2748/tmj/1178227573
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发表时间：
1990
影响因子：
0.5
通讯作者：
U. Rehmann
中科院分区：
文献类型：
--
作者：
Jun Morita;U. Rehmann
Introduction. Let A = (aij)1 <ij<n be an n x n generalized Cartan matrix, and g the Kac-Moody algebra over the field C of complex numbers, defined by A, with simple roots i7 = {αl5 , απ} and simple co-roots 77* = {α* |αe/7}, where we denote by α* the co-root of α (cf. [3]). Put α/?* = α(/J*). Associated to g and an arbitrary field F, we can construct a universal Kac-Moody group G(A, F), and the Steinberg group St(A, F). Let K2(A, F) be the kernel of the canonical homomorphism of St(A, F) onto G(A, F) (cf. Section 2). Matsumoto [4] has given a presentation of K2(A, F) if A is of finite type. As a natural generalization of his result, we will here give a presentation of K2(A, F) for arbitrary A. Let L be the abelian group generated by the symbols ca(u, v) for all α e Π and M, V in the multiplicative group F x of F with the following defining relations: (M1) cα(ί, u)ca(tu, v) = cα(ί, uv)cju, v) (M2) c α (l, l)=l (M3) φ9v) = φ-\v) (M4) cα(w, t;) = cα(w, (1 — u)v) with u φ 1 (M5) cJί^if^φ^Ό) (M6) cα̂ (ίw, t;) = cα^(ί, v)caβ(u, v) (M7) cβ/ϊ(ί, MI;) = caβ(t, u)caβ(t, v) for all oc,βeΠ with α # β and t,u,veF, where cα/,(u, ι;) = cα(w, ι; α *̂) = cβ{u *\ v). Then we obtain the following:
