A note on the exponential map of a real or \(p\)-adic Lie group
type: Item
instance of: scholarly article
title: A note on the exponential map of a real or \(p\)-adic Lie group
zbMATH Open document ID: 0846.22002
DOI: 10.1016/0022-4049(94)90123-6
author: Martin Moskowitz
author: Lawrence J. Corwin
published in: Journal of Pure and Applied Algebra
publication date: 1994-12-01T00:00:00Z
review text: A real (resp. \(p\)-adic) Lie group \(G\) is said to have the additive exponential property AEP if for all (resp. all small) elements \(X,Y\) of the Lie algebra \({\mathfrak g}\) of \(G\) the equation \(\exp X \cdot \exp Y = \exp (X + Y)\) implies \([X,Y] = 0\). The group \(G\) is said to have the multiplicative exponential property MEP if \(\exp X \cdot \exp Y = \exp Y \cdot \exp X\) implies \([X,Y] = 0\). The authors show that a connected real Lie group has AEP exactly if it is nilpotent and has the property that \([X,Y] \in {\mathfrak z} ({\mathfrak g})\) and \(\exp ({1 \over 2} [X,Y]) = 1\) imply \([X,Y] = 0\). Consequently, every simply connected (or, more generally, every linear) connected nilpotent real Lie group has AEP. In the \(p\)-adic case, the picture is quite different. For instance, it is proved that the group of \(\mathbb{Q}_p\)-rational points of any algebraic group defined over \(\mathbb{Q}_p\) has AEP on \({\mathfrak g}_0\), if only \(p > 3\). Here \({\mathfrak g}_0\) denotes the Lie ring of all ``small'' elements of \({\mathfrak g}\) (that is, those where a given valuation takes values greater than \((p - 1)^{-1})\). A connected nilpotent real Lie group has MEP exactly if it has AEP. However, there are nonnilpotent groups with MEP. Every connected real Lie group with MEP is solvable, and its adjoint representation has no purely imaginary roots. In the simply connected case, the latter condition implies the first. Finally, the authors prove that all \(p\)-adic linear groups have MEP.
Mathematics Subject Classification ID: 22E15
zbMATH DE Number: 699139
zbMATH Keywords: Lie group
zbMATH Keywords: additive exponential property
zbMATH Keywords: Lie algebra
zbMATH Keywords: multiplicative exponential property
zbMATH Keywords: valuation
zbMATH Keywords: adjoint representation
reviewed by: Markus Stroppel
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90123-6
OpenAlex ID: W2020206391
Wikidata QID: Q115364401
cites work: Q5339349
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description: scientific article; zbMATH DE number 699139