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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 \({\mathfrak g}\). Picking a basis in \({\mathfrak g}\), we obtain a group morphism Ad: \(G\to GL_ m{\mathbb{R}}\), 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 \(G^ X\) denote the group of all continuous maps \(X\to G\) and let G(A) denote the group of all \(g\in G^ X\) such that \(Ad(g)\in GL_ m(A)\). 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 \(l_ m\) 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 \(G(B)^ 0\) 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: \(a^{1/n}\in GL_ 1A[i]\) 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 \(a^{1/N}\) is defined by the series \[ a^{1/N}=(1+(a-1))^{1/N}=1+(a- 1)N+(a-1)^ 2(1/N)(1/N-1)+... \] Then for any simple Lie group G of classical type, a subgroup H of G(A) is normalized by \(G(A)^ 0\) if and only if \(G(B)^ 0\subset H\subset G(B)\) for an ideal B of A.
Reviewer: G.A.Margulis

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
20E07 Subgroup theorems; subgroup growth
20G35 Linear algebraic groups over adèles and other rings and schemes
20H25 Other matrix groups over rings
17B65 Infinite-dimensional Lie (super)algebras
17B20 Simple, semisimple, reductive (super)algebras
17B45 Lie algebras of linear algebraic groups

Citations:

Zbl 0658.00005