By Terence Tao
Winner of the 2015 Prose Award for most sensible arithmetic e-book! within the 5th of his well-known checklist of 23 difficulties, Hilbert requested if each topological crew which was once in the community Euclidean was once actually a Lie crew. in the course of the paintings of Gleason, Montgomery-Zippin, Yamabe, and others, this question was once solved affirmatively; extra often, a passable description of the (mesoscopic) constitution of in the community compact teams used to be demonstrated. to that end, this constitution concept used to be used to turn out Gromov's theorem on teams of polynomial progress, and extra lately within the paintings of Hrushovski, Breuillard, eco-friendly, and the writer at the constitution of approximate teams. during this graduate textual content, all of this fabric is gifted in a unified demeanour, beginning with the analytic structural concept of actual Lie teams and Lie algebras (emphasising the function of one-parameter teams and the Baker-Campbell-Hausdorff formula), then providing an evidence of the Gleason-Yamabe constitution theorem for in the community compact teams (emphasising the position of Gleason metrics), from which the answer to Hilbert's 5th challenge follows as a corollary. After reviewing a few model-theoretic preliminaries (most particularly the speculation of ultraproducts), the combinatorial purposes of the Gleason-Yamabe theorem to approximate teams and teams of polynomial progress are then given. a good number of suitable routines and different supplementary fabric also are supplied.
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Additional info for Hilbert's Fifth Problem and Related Topics
This intuition can be formalised fairly easily in the case of matrix Lie groups such as GLn (C); for more abstract Lie groups, one can still formalise things using nonstandard analysis, but we will not do so here. 8. (i) Show that the Lie algebra gln (C) of the general linear group GLn (C) can be identified with the space Mn (C) of n × n complex matrices, with the Lie bracket [A, B] := AB − BA. (ii) Describe the Lie algebra un (C) of the unitary group Un (C). (iii) Describe the Lie algebra sun (C) of the special unitary group SUn (C).
Hint: from (i) and (ii), one may identify g with the Lie algebra of a local linear Lie group. ) (iv) (Lie’s third theorem, simply connected version) Every finite-dimensional Lie algebra g is isomorphic to the Lie algebra of some global connected, simply connected Lie group. Furthermore, this Lie group is unique up to isomorphism. (v) Show that every local Lie group G has a neighbourhood of the identity that is isomorphic to a neighbourhood of the identity of a global connected, simply connected Lie group.
Let G be a Lie group with Lie algebra g. For any X, Y ∈ g, show that 2 exp([X, Y ]) = lim (exp(X/n) exp(Y /n) exp(−X/n) exp(−Y /n))n . 3. Building Lie structure from representations and metrics Hilbert’s fifth problem concerns the minimal hypotheses one needs to place on a topological group G to ensure that it is actually a Lie group. 2, we saw that one could reduce the regularity hypothesis imposed on G to a “C 1,1 ” condition, namely that there was an open neighbourhood of G that was isomorphic (as a local group) to an open subset V of a Euclidean space Rd with identity element 0, and with group operation ∗ obeying the asymptotic x ∗ y = x + y + O(|x||y|) for sufficiently small x, y.