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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 $\frac{n(n-1)}{2}.$

7686 questions
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Using Lie theory to solve a problem in SE(2)

For background, I have little to no experience with Lie theory but from what I do know it feels like this problem could be easily solved using it, so I'm curious to see how I can get to the solution using the tools provided by Lie…
japata
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About a generalisation of Raghunathan's theorem

The Raghunathan's theorem states that if $G$ is a simple Lie group non-compact and $g:G\to GL_n(\mathbb{R})$ is a linear representation without fixed point and $\Gamma$ a discrete co-compact subgroup then, $H^1(\Gamma,\mathbb{R}^n)=0$ with…
théo jamin
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Universal cover of $SO^+(p,q)$

In the Wikipedia page of $SO^+(p,q)$, they do not mention any details about its universal cover. Google searches also did not give any relevant results. So, what is the universal cover of $SO^+(p,q)$, and how do we prove that it is indeed the…
Ishan Deo
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Linear Lie groups - exp of a differentiable morphism

I am doing some selfstudy on Linear Lie groups and trying do an exercise. Let G be a linear Lie group and $\phi$ a differentiable morphism from GL(n,$\mathbb{R}$) into G. Define $\Phi = (D_\phi)_I$. a) show that for every $X \in M(n,\mathbb{R})$…
user622405
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Why are all $\mathbb{R}$-actions on compact lie groups given by conjugations?

Let $G$ be a compact lie group and $\alpha:\mathbb{R}\to \text{Aut}(G)$ be a group homomorphism with continuous orbit maps $t\mapsto \alpha_t(g)$, for $g\in G$. Why is there always an $X\in \textbf{L}(G)$ such that $\alpha_t(g)=\text{exp}(tX)\cdot…
Tobias Simon
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Left invariance on m-forms

Given a Lie Group $G$ of dimention $n$, how can we show that left invariant $m$-forms form a vector space of dimension $\binom{n}{m}$? I know that, for a vector field of dimension $n$, these $m$-forms form a vector space of dimension $\binom{n}{m}$…
MicrosoftBruh
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$\phi(A) =(I_n-A)(I_n+A)^{-1}$ is antisymmetric?

Let $U=\{A\in SO_n(\mathbb R)/ \;\det(I_n+A)\neq0\}$. Let $\phi :U\to M_n(\mathbb R)$ the application defined by : $$\phi(A)=(I_n-A)(I_n+A)^{-1}$$ How can I show that $\phi(A)$ is antisymmetric !? Any help is highlty appreciated!
M-S
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path component and orbits

Consider $G$ lie group and $M$ riemannian manifold, what does it mean to say that that the points in M/G corresponding to the orbit G(p) and G(q) are in the same path component of M/G?
Toy
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one-parameter subgroup of a one-dimensional connected Lie group

Let $G$ be a connected Lie group of dimension one and let $U=\{u_t\}_{t \in \mathbb R }$ be a one-parameter subgroup. I wonder if it is true that $U=G$? It is easy to see that $U$ is closed. So by a theorem in Lie theory, $U$ is a Lie subgroup. $U$…
No One
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Topological group admits up to isomorphism only one smooth structure turning it into a Lie group.

proposition : Let $G$ be a Lie group. Then $G$ as a topological group admits, up to isomorphism, only one differentiable structure turning it into a Lie group. I think I have to use the fact that if $f : G\to H$ is a continuous homomorphism…
roi_saumon
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Show that Stabilizer of Lie group action is closed lie subgroup

Let $G$ be a Lie group and $M$ a smooth manifold on which $G$ acts. For $x \in M$, how do I show that $G_{x} = \{ g \in G : g \cdot x = x\}$ is a closed lie subgroup? What I was thinking is that $f : G \times M \rightarrow M$ defined by $(g,m)…
user100101212
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Closed Lie subgroups

Suppose that $\phi : G \rightarrow H$ is a lie group homomorphism. How do I show that $\ker(\phi)$ is a closed subgroup of $G$? In general, when we refer to $H \leq G$ being closed, do we mean that $H$ is closed under the topology associated with…
user100101212
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Can a simply-connected Lie group have a non-trivial central extension by a discrete group?

(1) Suppose a (finite-dimensional, real) Lie group $G$ is connected and simply-connected. Can $G$ have a non-trivial central extension by a discrete group, i.e. can there exist a Lie group $\tilde G$ such that $G \simeq \tilde G/Z$ where $Z$ is a…
user152767
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On the stabilizer

Let $A$ be $n\times n$ positive-definite matrix over $\mathbb{R}$. Does $\lbrace X\in M_{n\times n}(\mathbb{R})|X^TAX=A\rbrace\cong O(n;\mathbb{R})$ (homeomorphic)?
Topologieeeee
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Where is the inverse function theorem being used here?

I am posting a picture, containing a theorem and its proof from these notes on Lie groups. I'm quoting one of the lines: $UH$ is open in $G$ (which easily follows from inverse function theorem applied to the map $f:U\times H\to G$). What…
user67803
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