---- > [!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 $\mathsf{Lie}$. > Categorically speaking, a [[Lie group]] is precisely a [[group|group object]] in $\mathsf{Man}^{\infty}$. table below can add stuff to | If you hear... | Think | | ------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [[Compact]] $\mathsf{CompLie}$ | [[compact group representations are unitary]], work with an [[unitary group representation\|invariant inner product]], [[compact covering criterion for smooth manifolds]] (but usually have access to better for Lie groups, see below) | | [[simply connected]] $\mathsf{LieSC}$ | [[the category of simply connected Lie groups is equivalent to that of finite-dimensional real lie algebras]], enough to consider the Lie algebra | | [[connected]] $\mathsf{Con nLie}$ | [[connected topological group is generated by any neighborhood of the identity\|generated by any neighborhood of]] $e$ (+corollary), [[on the universal cover of a connected Lie group]] (e.g. is a quotient of a [[simply connected]] [[Lie group]]), [[Lie group homomorphisms with discrete kernels]], [[the Lie subgroup–subalgebra correspondence]], [[homomorphism out of a connected Lie group is determined by its induced Lie algebra homomorphism]] | | [[semisimple Lie group\|semisimple]] | [[The Cartan-Killing Criterion]] | | [[simple group\|simple]] | [[classification of complex semisimple Lie algebras\|Cartan-Killing classification]] | | [[abelian group\|abelian]] + [[connected]] | [[classification of connected abelian Lie groups]], $\exp$ is a LGH | | | | Also, try to think with matrix Lie groups first and then reformulate what you've said with $\operatorname{Ad}$/[[Lie exponential map|exponential map]]/etc. > [!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 > [!basicproperties] > - The [[kernel of a group homomorphism|kernel]] $\operatorname{ker }\varphi$ of a Lie group homomorphism $\varphi:G \to H$ is a [[closed set|closed]], [[closed subgroup theorem|hence]] [[embedded submanifold|embedded]], [[subgroup]] of $G$. (It is the [[inverse image]] of a [[closed set|closed set]] in $H$.) > - Lie groups are [[topological group|topological groups]], and hence inherit properties like [[connected topological group is generated by any neighborhood of the identity]]. As a corollary, get things like [[homomorphism out of a connected Lie group is determined by its induced Lie algebra homomorphism]]. ^properties WIP: [[Lie group table]], [[Lie groups.canvas]] > [!basicexample] > The [[conjugate|conjugation]] ([[inner automorphism]]) $x \xmapsto{C_{g}}gx g^{-1}$ 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 Where is a good place to collect examples like $\mathbb{S}^{1}$ (and $\mathbb{S}^{3}$), torus? [[general orthogonal group|orthogonal]] [[the orthogonal group is a Lie group|group]] [[special orthogonal group]] [[general orthogonal group|unitary group]] (todo bring over lg sheet 1) [[the special unitary group is a Lie group|special unitary group]] [[the special linear group is a Lie group|special linear]] [[special linear group|group]] ---- #### ---- #### 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 > ```