---- > [!definition] Definition. ([[Lie group]]) > A **Lie group** is a [[group]] $G$ that is also a finite-dimensional real [[smooth manifold]], in which the group operations of multiplication $G \times G \to G$ and inversion $G \to G$ are [[smooth maps between manifolds|smooth]]. > > A **Lie group homomorphism $G \to H$** is a [[smooth maps between manifolds|smooth]] map $G \to H$ that is also a [[group homomorphism]]. The [[category]] of [[Lie group|Lie groups]] and their homomorphisms is denoted $\mathsf{Lie}$. ^definition > [!note] Remark. > It is enough to check that the mapping $\begin{align} G \times G \to& G \\ (x,y) \mapsto& x ^{-1} y \end{align}$ is a [[smooth maps between manifolds|smoothing mapping]] from the product manifold into $G$. ^note > [!basicexample] > The [[conjugate|conjugation]] ([[inner automorphism]]) $x \xmapsto{C_{g}}x g x ^{-1}$ is a Lie group [[group isomorphism|isomorphism]] $G \to G$ for any $g \in G$. Left-translation $x \xmapsto{L_{g}} gx$ is not; neither is right-translation $R_{g}$. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```