Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSimplified Proofs for the Pro-Lie Group Theorem and the One-Parameter Subgroup Lifting Lemma
142
0
0.0
(
0
)
Authors
Helge Glockner
Ask ChatGPT about the research
No Arabic abstract
This note is devoted to the theory of projective limits of finite-dimensional Lie groups, as developed in the recent monograph ``The Lie Theory of Connected Pro-Lie Groups by K.H. Hofmann and S.A. Morris. We replace the original, highly non-trivial proof of the One-Parameter Subgroup Lifting Lemma given in the monograph by a shorter and more elementary argument. Furthermore, we shorten (and correct) the proof of the so-called Pro-Lie Group Theorem, which asserts that pro-Lie groups and projective limits of Lie groups coincide.
rate research
Read More
A weakly complete vector space over $mathbb{K}=mathbb{R}$ or $mathbb{K}=mathbb{C}$ is isomorphic to $mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is illustrated by the fact that the underlying vector space of the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly complete real or complex associative algebras are studied because they are necessarily projective limits of finite dimensional algebras. The group of units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the associated topological Lie algebra $A_{rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $expcolon Ato A^{-1}$ as the exponential function of $A^{-1}$. With each topological group, a weakly complete group algebra $mathbb{K}[G]$ is associated functorially so that the functor $Gmapsto mathbb{K}[G]$ is left adjoint to $Amapsto A^{-1}$. The group algebra $mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ is compact, the $mathbb{R}[G]$ contains $G$ as the set of grouplike elements. The category of all real Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is shown to be equivalent to the category of compact groups. The group algebra $A=mathbb{R}[G]$ of a compact group $G$ contains a copy of the Lie algebra $mathcal{L}(G)$ in $A_{rm Lie}$; it also contains a copy of the Radon measure algebra $M(G,mathbb{R})$. The dual of the group algebra $mathbb{R}[G]$ is the Hopf algebra ${mathcal R}(G,mathbb{R})$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis of a duality ${mathcal R}(G,mathbb{R})leftrightarrow mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality.
A valuated group with normal forms is a group with an integer-valued length function satisfying some Lyndons axioms and an additional axiom considered by Hurley. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko-Neumanns theorem. We study also the CSA property in such groups.
The goal of this paper is to give a group-theoretic proof of the congruence subgroup property for $Aut(F_2)$, the group of automorphisms of a free group on two generators. This result was first proved by Asada using techniques from anabelian geometry, and our proof is, to a large extent, a translation of Asadas proof into group-theoretic language. This translation enables us to simplify many parts of Asadas original argument and prove a quantitative version of the congruence subgroup property for $Aut(F_2)$.
We study the class of finite groups $G$ satisfying $Phi (G/N)= Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
We show that for every finitely generated closed subgroup $K$ of a non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$, and a continuous homomorphism $tau colon U to K$ satisfying $tau(k) = k$ for every $k in K$. We prove that the intersection of a pair of finitely generated closed subgroups of a Demushkin group is finitely generated (giving an explicit bound on the number of generators). Furthermore, we show that these properties of Demushkin groups are preserved under free pro-$p$ products, and deduce that Howsons theorem holds for the Sylow subgroups of the absolute Galois group of a number field. Finally, we confirm two conjectures of Ribes, thus classifying the finitely generated pro-$p$ M. Hall groups.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
