# Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich

By A. I. Kostrikin, I. R. Shafarevich

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Van Kampen [125] in the early thirties of the 20th century. For any topological abelian group G we let G = Hom(G, T) denote its dual with the compact open topology. g. ) There is a natural morphism of abelian groups ηG : G → G given by ηG (g)(χ) = χ (g) which may or may not be continuous; information regarding this issue is to be found for instance in [102, pp. 7 on p. 300. We shall call a topological abelian group semireflexive if ηG : G → G is bijective and reflexive if ηG is an isomorphism of topological groups; in the latter case G is also said to have duality (see [102, p.

44). Let G be a finite-dimensional connected pro-Lie group with Lie algebra g. Then G is locally compact metric, and there is a compact metric totally def disconnected member ∈ N (G) such that the Lie group F = G/ and the quotient morphism ρ : G → F satisfy the following conditions: (∗) There is a morphism ϕ : F → G such that πF = ρ ϕ. (∗∗) = ϕ(P (F )). def (∗∗∗) Let D = {(ϕ(g)−1 , g) : g ∈ P (F )} ∼ = P (F ); then ×F →G D is a well-defined isomorphism of locally compact metric groups, and × F , × F and G are all locally isomorphic.

We shall see that this is largely the case for pro-Lie algebras due to the fact that the underlying topological vector spaces are weakly complete vector spaces and that these have a perfect duality theory that allows us to translate their topological linear algebra to pure linear algebra upon passing to the vector space duals. ) The Module Theory of Pro-Lie Algebras We saw that for every pro-Lie group G there exists a simply connected pro-Lie group G and a natural morphism with dense image πG : G → G.