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 ...

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Cohomology ring of $U(n)$

As you know $$H^\ast (U(n);{\bf Z})=\bigwedge_{\bf Z}[x_1,x_3,...,x_{2n-1}]$$ where $|x_i|=i$ To prove this we use Leray-Hirsch Theorem for $$\tag{*}\ U(n-1)\rightarrow U(n)\rightarrow S^{2n-1}$$ ...
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265 views

Is the Lie bracket of a Lie Algebra of a Matrix Lie Group always the commutator?

Given a Matrix Lie Group, the Lie Bracket is of the associated Lie Algebra is given by the Lie Derivative. Is this always the commutator if we start from a Matrix Lie Group? Cheers!
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432 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
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2answers
324 views

Learning representation theory of Lie groups for someone who knows Lie algebras

I'd like to learn the representation theory of Lie groups. I have a good knowledge of semisimple Lie algebras and their representation theory as well as the basics of Lie groups. To what extent are ...
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152 views

Is the additive group of real numbers (R,+) compact?

I have a very naive question about basic topology. My goal is to determine the conditions on a vector field in order for its flow to define a proper (R,+)-action. I understand that if the group G is ...
7
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1answer
148 views

Is taking inverse automatically well-defined?

By the usual definition, Lie group is a manifold $G$ with a group structure on it such that the multiplication $m\colon G\times G\to G$ and taking inverse $i\colon G\to G$ are both smooth maps. But it ...
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53 views

Exponential intertwining of linear and local actions.

I am reading Duistermaat's 1973 paper on relating the convexity of the image of a moment map to the image of the fixed points of an antisymplectic involution. In that paper, the following comment is ...
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1answer
385 views

Lie algebra of normal subgroup is an ideal

I want to prove that if $G$ is a connected Lie group, $H$ is a normal Lie subgroup of $G$, $\mathfrak{g}$ and $\mathfrak{h}$ their respective lie algebras, then $\mathfrak{h}$ is an ideal of ...
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468 views

Computing Sylow $p$-subgroups of classical groups

Let $p>4$ be prime, and let $G=GL_2(\mathbb{F}_p)$, $H=O_3(\mathbb{F}_p)$, and $K=Sp_4(\mathbb{F}_p)$. We know that $|G|=p(p-1)^2(p+1)$, so that a Sylow $p$-subgroup of $G$ is isomorphic to ...
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77 views

Does the Lie Bracket automatically exist?

Let $g$ be a Matrix Lie Group. The Lie Algebra of $g := Lie(g)$ is defined as $ Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \} $ ...
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138 views

When are finite-index subgroups of a Lie group closed?

Let $G$ be a Lie group (or, if necessary, a reductive Lie group) and $H$ a subgroup of $G$. If $\lbrack G:H\rbrack < \infty$, is it true that $H$ is closed? If not, are there any broad assumptions ...
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60 views

A question about double cover of Lie group

If the fundamental group of a symplrctic Lie group be infinite cyclic, why it should has a unique connected double cover?
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156 views

Quaternionic general linear group is open

Is there an elegant proof of the following fact: "The quaternionic general linear group $GL(n, \mathbb{H})$ is open in $M_n(\mathbb{H})$", where $M_n(\mathbb{H})$ is the set of all $n \times n$ square ...
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70 views

Calculating an expression for the trace of generators of two Lie algebra.

Suppose we have $$[Q^a,Q^b]=if^c_{ab}Q^c$$ where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have $$[P^a,P^b]=0$$ where P's are generators of a Lie ...
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67 views

Prove $\exists$ neighborhood of $I \in Gl(n,\mathbb{C})$ containing no nontrivial subgroup.

Prove that there exists a neighborhood of the identity $I \in Gl(n,\mathbb{C})$ that contains no subgroup other than $\left\{ I \right\}$. Thanks!
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92 views

Geometry of Lie Group Around Identity

Let $G$ be a continuous compact Lie group. And let $K,\ H$ be closed subgroups. How can we take $W$ which is a small open set around $e$ and satisfies the following : If $K\subset WH$ then $KH/H$ ...
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100 views

Is Bruhat cell dense in p-adic topology?

I've seen in literature a statement like 'there exists an open and dense Bruhat cell'. In $GL(2,F)$ for example, where $F$ is a p-adic field, let $\omega=\begin{pmatrix} & 1 \\ 1 & ...
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1answer
60 views

The tangent space of $\mathrm{Aut}(T_eG)$

Let $G$ be a Lie group and $e \in G$ be the identity. I want to understand the following sentence. " $\mathrm{Aut}(T_eG)$ being just an open subset of the vector space of endomorphisms of $T_eG$, its ...
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56 views

Metric over a Lie algebra $\mathfrak{u}(n)$

Let $\mathfrak{u}(n)$ be the Lie algebra of the Lie group $U(n)$. I can define a positive-definite inner product over $\mathfrak{u}(n)$ in this way: if $A,B \in \mathfrak{u}(n)$ I define $\langle A,B ...
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109 views

Prove $G = \left\{ \mathrm{diag} (e^{ti}, e^{\lambda ti}) \mid t \in \mathbb{R} \right\}$ is not a manifold.

Let $\lambda$ be an irrational number. Let $G \subset G_2(\mathbb{C})$ be defined as $G = \left\{ \mathrm{diag} (e^{ti}, e^{\lambda ti}) \mid t \in \mathbb{R} \right\}$. Prove that $G$ is not a ...
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89 views

Momentum map and equivariance

I am reading an article in which I do not understand some equivariance property about the momentum map. Let $G$ be a Lie group acting on a manifold $Q$. The action is denoted $(g,q) \, \mapsto \, q ...
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Representing $\mathbb{R}/\mathbb{Z}$ as a matrix group.

It was told to me that $G = \mathbb{R}/\mathbb{Z}$ is a real matrix group. Can someone help me understand how to represent $G$ in $Gl_n(\mathbb{R})$ for some $n$? (Supposedly, $n = 1$? But that's ...
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1answer
132 views

Finding the lie algebra of the symplectic lie group

I am having difficulties completing my proof that $\text{Lie}(\text{Sp}(2n)) \equiv \mathfrak{sp}(2n) = \{ X \in Gl(2n)\; |\; X^TJ + JX = 0 \}$ Where $J \equiv \begin{bmatrix}0 & \mathbb{1}_n ...
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1answer
45 views

Does $\mathrm{Im}(\exp)$ being a manifold imply the domain is a manifold?

Let $G$ be a real matrix group of dimension $n$. Let $\mathfrak{g}$ denote the lie algebra of $G$. Suppose $X \subset \mathfrak{g}$ such that $e^X$ is a submanifold of $G$ of dimension $m$. Does ...
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43 views

Does every element of $G_2 = \mathrm{Aut}(\mathbb{O})$ stabilize a quaternion subalgebra?

Has anyone heard of this result before? Let $\mathbb{O}$ denote the octonions and let $G_2$ denote its automorphism group (i.e. the 14 dimensional subgroup of $SO(7)$). Then any element of $G_2$ ...
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316 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
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100 views

Which Lie group / algebra is generated by these three matrices?

This is a beginner question (and not any homework). I want to get a feeling for Lie group/algebra generators. Do the three matrices $$A=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0& ...
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1answer
760 views

What are defining & fundamental representations?

In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?
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282 views

Conditions for a topological group to be a Lie group.

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157): Let $G$ be a locally compact ...
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63 views

Does smooth section of a quotient space $G/H$ define an immersion?

Question 1: Let $G$ be a Lie group and $H<G$ a Lie subgroup of $G$, given the projection $p:G\to G/H$ and a smooth (local) section $s:G/H\to G$ s.t. $p\circ s=id$, then does this imply that $s$ is ...
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1answer
511 views

Expression for the Maurer-Cartan form of a matrix group

I understand the definition of the Maurer-Cartan form on a general Lie group $G$, defined as $\theta_g = (L_{g^{-1}})_*:T_gG \rightarrow T_eG=\mathfrak{g}$. What I don't understand is the expression ...
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1answer
113 views

lagrangian subspace and Heisenberg group

Let $(V,\omega)$ be a symplectic vector space. Also we assume $L\subset V$ be a Lagrangian subspace., and $H(V)$ be Heisenberg group, then why $L\bigoplus U(1)\subset H(V)$ is maximal abelian ...
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99 views

Characteristic functions of group-invariant probability distributions

Suppose that we have a probability distribution $\rho(\mathbf x)$ on a manifold $\mathcal M$, which is invariant under the action of a Lie group $G$, $\rho(g\mathbf x)=\rho(\mathbf x)$ for all ...
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119 views

Contractibility vs. G-contractibility

Let $X$ be a space equipped with an action of a compact Lie group $G$. Recall that such a space is said to be $G$-contractible if the identity map of $X$ is $G$-homotopic (i.e., homotopic through ...
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73 views

Measurable Homomorphism between $\Bbb R$ and Lie Groups

Let φ : $f:\Bbb R \to G $ be a measurable homomorphism, where $G$ is a Lie Group with a choice of Haar measure. How can I prove that $\varphi$ is continuous? When $ G=\Bbb R$ is well known that ...
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68 views

What is the Weyl group of this group?

Let $G$ be the group $GL_{n_1}(q^{l_1})\times GL_{n_2}(q^{l_2})\times GL_{n_3}(q^{l_3})$. Here $GL_{n_1}(q^{l_1})$ is the rational points of $GL(n,\bar {\mathbf{F}}_q)$. My question iis what is the ...
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208 views

What is group manifold of a compact Lie Group?

I searched on google the meaning of a group manifold of a compact lie group but I didn't get the answer. A paper on arxiv "Background Independent Quantum Gravity:A Status Report- Abhay Ashtekar" on ...
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40 views

Topology of Lie groups. [duplicate]

I do not understand the topology of a Lie group clearly. Let $G$ be a Lie group and $T_eG$ be its tangent space at the identity $e \in G$. Why $Aut(T_eG)$ is an open subset of the vector space of ...
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1answer
93 views

Proofs that: $\text{Sp}(2n,\mathbb{C})$ is Lie Group and $\text{sp}(2n,\mathbb{C})$ is Lie Algebra

Consider following Lie Group: $$ \text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}(\mathbb{C})\mid J=g^TJg\}\quad\ where\quad J=\begin{pmatrix} 0 & 1_n \\ -1_n & 0 \end{pmatrix} $$ And the ...
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1answer
56 views

Determine a basis for the Lie-Algebra $\text{sp}(\text{2n},\mathbb{C})$

Consider the Lie Group $\text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}\mid\ J=g^TJg\}$ where $J=\begin{pmatrix} 0 & 1_n \\ -1_n & 0 \end{pmatrix} $. The corresponding Lie Algebra is ...
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218 views

Dimension of isometry group of complete connected Riemannian manifold

Given an $n$-dimensional geodesically complete connected Riemannian manifold $M$, we want to prove that the dimension of its isometry group is $$\dim {\rm ISO}(M) \leq \frac{n(n+1)}2.$$ Does it ...
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152 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
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1answer
44 views

relation between the Poincaré and Euclidean algebra

Take $d$ a strictly positive integer, and consider the (proper) Euclidean group $E^d$ (the symmetry group of $\mathbf{R}^d$ with the conventional inner product), and the (proper, ortochronous) ...
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46 views

Transitive topological action

Let $G$ be a topological group, $A$ be set and $\mu\colon G\times A\to A$ a transitive action. I'm trying to prove the statement below is false. There exists only one topology in $A$ such ...
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43 views

Orthochronous Lorentz is time preserving and $\operatorname{SL}(2,R)$

Let's consider the pseudosphere in $\mathbb{R}^{1,2}$ given by $x^2+y^2-z^2=-R^2.$ We know that the group $O(1,2)=\{ A \in \operatorname{Mat}(3,\mathbb{R}): A^tGA=G \}$ where ...
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94 views

$\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group?

I try to prove $\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group. My idea is that since $\exp\colon \mathfrak{sl}(2,\mathbb{R}) \to ...
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127 views

decomposition of right Haar measure on homogeneous space

For simplicity, let $G_n=GL(n,\mathbb{R})$, $N_n$ be the upper trianguler unipotent subgroup, $P_{n-i}$ be the standard parabolic subgroup associated to partition $n=(n-i,i)$, and finally let $K=O(n)$ ...
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750 views

Ergodic action of a group

What does it mean and how is it defined if the action of a group is meant to be ergodic? Thank you for your replies!
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394 views

References for basic level Differentiable Manifolds and Lie Groups

I an undergraduate math student with a decent background in abstract algebra. I am looking forward to studying Lie groups this summer...I want some you to suggest good references for the following ...
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102 views

Asymptotic invariants of infinite groups

I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept : $$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$ That is for ...