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Normal subgroups of gauge groups. (English) Zbl 0679.22013
Classical groups and related topics, Proc. Conf., Beijing/China 1987, Contemp. Math. 82, 199-220 (1989).
[For the entire collection see Zbl 0658.00005.] Let G be a simple Lie group and let Ad denote the adjoint representation of G on its Lie algebra ${\frak g}$. Picking a basis in ${\frak g}$, we obtain a group morphism Ad: $G\to GL\sb m{\bbfR}$, where $m=\dim (G)$. Let X be a topological space and A be a ring of bounded real continuous functions on X containing all constants. Let $G\sp X$ denote the group of all continuous maps $X\to G$ and let G(A) denote the group of all $g\in G\sp X$ such that $Ad(g)\in GL\sb m(A)$. For any ideal B of A, let G(B) denote the group of all g in G(A) such that Ad(g) is congruent to the identity matrix $l\sb m$ modulo B. Let us endow A with the topology of uniform convergence, and consider the induced topology on G(A). For every ideal B of A, let $G(B)\sp 0$ denote the connected component of identity in G(B). The main result of the paper is the following Theorem. Let X be a topological space, and let A be a ring of bounded real continuous functions on X containing all constants and satisfying the following condition for all natural numbers N: $a\sp{1/n}\in GL\sb 1A[i]$ for any $a\in A[i]$ sufficiently closed to 1 where A[i] is the subring of all complex functions a on X with both real and imaginary part in A, and $a\sp{1/N}$ is defined by the series $$ a\sp{1/N}=(1+(a-1))\sp{1/N}=1+(a- 1)N+(a-1)\sp 2(1/N)(1/N-1)+... $$ Then for any simple Lie group G of classical type, a subgroup H of G(A) is normalized by $G(A)\sp 0$ if and only if $G(B)\sp 0\subset H\subset G(B)$ for an ideal B of A.
Reviewer: G.A.Margulis

MSC:
22E65Infinite-dimensional Lie groups and their Lie algebras: general properties
20E07Subgroup theorems; subgroup growth
20G35Linear algebraic groups over adèles and other rings and schemes
20H25Other matrix groups over rings
17B65Infinite-dimensional Lie (super)algebras
17B20Simple, semisimple, reductive Lie (super)algebras
17B45Lie algebras of linear algebraic groups