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Questions tagged [lie-groups]

A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group.

Consider using with the (group-theory) tag.

Lie groups are groups that are also differentiable manifolds that represent the best developed theory of continuous symmetry of mathematical objects.

Examples of lie groups are:

1) The Euclidean space $\mathbb{R}^n$ under addition is a lie group.

2) The special orthogonal group of real orthogonal matrices with determinant $1$ (note that $n=3$ is the rotation group in $\mathbb{R}^3$).

3) The spin group, which is the double cover of the special orthogonal group such that $\exists$ a sequence of lie groups:

\begin{equation*} 1\to Z_2\to~\text{Spin}(n)\to SO(n)\to 1. \end{equation*}

Note that it has dimension $\frac{n(n-1)}{2}.$

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Non injectivity of the Lie group exponential

Let $G$ be a finite dimensional Lie group. Suppose there is a point $g$ such that there exists two tangent vectors $X,Y\in T_eG$ with $X\neq Y$ and $\exp(X)=exp(Y)=g$. In other words, the group exponential is not injective. Does it tell us something…
Chevallier
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Motivation for the exponential in the definition of and element of a Lie group?

Can I get a spoiler on where this definition of the elements of a Lie group is headed? In this lecture Alex Flournoy does a great job at introducing the elements of a Lie group. However, there is ultimately this unmotivated definition: A general…
Antoni Parellada
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A technical detail in the definition of a Lie group

I begin with the following common Definition: A Lie group $ G $ is a differentiable (smooth, analytic) manifold equipped with a group structure, i.e. with an associative binary operation $$ G\times G \longrightarrow G :\qquad\left\{x\,, y\right\}…
Michael_1812
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Can the projective unitary group be realized as a matrix Lie group

Is the projective unitary group $U(n)/U(1)$ isomorphic to some closed subgroup of the $GL(n,\mathbb{C})$ (i.e., a matrix Lie group)?
Andrew Yuan
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Why ${\rm Lie}(H)=\{X \in {\rm Lie}(G):\exp(tX) \in H\}$?

I encountered a theorem about Lie algebra of Lie subgroups. It reads (for $G$ a Lie group and $H$ a Lie subgroup) : ${\rm Lie}(H)=\{X \in {\rm Lie}(G):\exp(tX) \in H\}$. First, I don't really understand this intuitively. Second, when I think about…
roi_saumon
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The closed subgroup of Lie group

$G$ is a connected Lie group with Lie algebra $g$ and $l$ is an abelian ideal of $g$. If $K$ is the connected Lie subgroup of $G$ with the Lie algebra $l$, then is $K$ necessarily closed in $G$?
Summer
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Quotient of a Lie group by normal lie subgroup is a lie group

If $G$ is a Lie group and $H \leq G$ is a closed normal lie subgroup, how do I show that $G/H$ is a lie group. I know that $f : G \times G \rightarrow G$, by $(g,h) \mapsto gh$ and $h : G \rightarrow G$ by $h \mapsto h^{-1}$ are smooth maps. How do…
user100101212
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generating a path-connected group from a neighborhood of 1

I am reading the book 'Naive Lie theory'. In it, it is proven that a path-connected group can be generated by a neighborhood of the unity element. The idea is simple and clear. But I cannot overcome some details. How is always possible to find…
John
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Is there a Lie group with compact elements?

It is possible that for a (non-abelian) Lie group, the subgroup generated by each element is compact?
user549766
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Linear Lie group.

Suppose that G is a connected Lie group such that the center of G is trivial. Question: Is it true that G is isomorphic (as Lie group) to a closed subgroup of a Linear group GL(n,R) for some natural number n. Or maybe there is an evident…
Ofra
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Conjugacy classes of the Lie group $Sp(6,\mathbb{R})$

I am trying to find the number of conjugacy classes of the Lie group $Sp(6,\mathbb{R})$, and identify which elements each of them contains (in matrix form). When searching literature on the topic, I find lots of examples where the conjugacy classes…
Calimero
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Non-trivial open connected subgroup of a connected Lie Group

Let $G$ be a connected Lie group. Then, I would like to show that there does not exist a open connected Lie subgroup $H$ such that $\{e\}\subsetneq H \subsetneq G$. Any help or hint would be very helpful!
Shubham Namdeo
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Closed Subgroup of $GL(n,\mathbb{K})$ is Lie group.

I am recently doing self-study for Lie Group. I am using Hall’s Elementary Introduction. In this book, it says the matrix Lie group is any subgroup $G$ of $GL(n,\mathbb{C})$, for any convergent sequence in $G$ it will converge to an element A such…
Hamio Jiang
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Tangent map of homomorphism of Lie groups preserves commutator

I am trying to prove the following proposition: Let $\phi:G\rightarrow H$ be a homomorphism of Lie groups, and $X,Y\in T_eG$. Then, $$ T_e\phi([X,Y]_G) = [T_e\phi X,T_e\phi Y]_H $$ So we want to prove $ T_e\phi\circ ad^G(X)(Y) = ad^{H}(T_e\phi…
soap
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Is it true that if a Lie group act trivially on an open subset of a manifold the action of the group is trivial (on the whole manifold)?

Of course, I assume the manifold is connected. I feel like this is probably true but I have no idea how to prove it. Any hints?
R Mary
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