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Normal subgroups of classical groups over rings and gauge groups. (English) Zbl 0676.20025
Algebraic K-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 451-459 (1989).
[For the entire collection see Zbl 0655.00010.] For any associative ring A with 1 and a natural number n, let $GL\sb nA$ be the group of all invertible n by n matrices over A and $E\sb nA$ the subgroup generated by all elementary matrices over A. For any ideal B of A, let $G\sb n(A,B)$ be the inverse image of the center of $GL\sb n(A/B)$ under the canonical homomorphism, $E\sb nB$ the subgroup of $E\sb nA$ generated by elementary matrices in $G\sb n(A,B)$ and $E\sb n(A,B)$ the normal subgroup of $E\sb nA$ generated by $E\sb nB$. The author discusses under what condition on A and n, the following hold: (1) A subgroup H of $GL\sb n(A)$ is normalized by $E\sb nA$ if and only if $E\sb n(A,B)\subset H\subset G\sb n(A,B)$ for an ideal B of A; (2) $E\sb n(A,B)=[E\sb nA,E\sb nB]=[E\sb n(A,B),GL\sb nA]=[E\sb nA,G\sb n(A,B)]$ for any ideal B of A. When $n\ge 3$, the statement has been proved for various classes of rings A including: rings which are finitely generated modules over their centers, von Neumann regular rings, Banach algebras. Partial results were obtained also for $n=2$ (in this case (1) and (2) are not always true). The author discusses also the analogous problem for the ordinary and pseudo orthogonal groups in general setting. He announces some new results on the statements analogous to (1) and (2) but the answer and proofs become more complicated, involving (when $2A=A)$ “quasi-ideals” of A rather than ordinary ideals even in the local ring case. He also discusses the analogous problem for infinite dimensional groups including loop groups.
Reviewer: E.Abe

MSC:
20G35Linear algebraic groups over adèles and other rings and schemes
20H25Other matrix groups over rings
20E07Subgroup theorems; subgroup growth
18F25Algebraic $K$-theory and $L$-theory
22E65Infinite-dimensional Lie groups and their Lie algebras: general properties