Lie group

李群

Geometry & Topology · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A group equipped with a finite-dimensional Hausdorff, second countable smooth manifold structure without boundary for which multiplication and inversion are smooth maps.

具有有限维、Hausdorff、第二可数的无边光滑流形结构，且乘法与求逆均为光滑映射的群。

Example

$S^1$ is a Lie group under complex multiplication; $\operatorname{GL}(n,\mathbb R)$ is one under matrix multiplication.

Usage and distinctions

Connectedness and compactness are additional properties. Its Lie algebra describes infinitesimal group structure but does not distinguish every pair of globally different Lie groups.

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Definition sources

1. Eckhard Meinrenken — Riemannian Geometry§10.1, Definition 10.1 and Proposition 10.2; manifold conventions in Definition 1.5 · Checked 2026-10-15

Translation sources

1. Fudan University — Sun Binyong, Lie Group short course课名：李群和李代数；课程内容：李群及例子、李群的李代数、指数映射 · Checked 2026-10-15

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