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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12 views

Liegroup acting transitively => connected component acts locally transitive?

Suppose $G$ is a Liegroup acting transitively on a manifold $M$. Does that already imply, that the connected component $G^\circ$ of $G$ acts transitively on the connected components of $M$? I have a ...
-2
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1answer
13 views

algebra operations

If $[x]=x^{i}\sigma^{i}$, Find an alternative form for the product $$ x^{i}n^{j}\sigma^{k}\epsilon^{ijk}=x^{i}(\vec{n} \times \vec{\sigma})^{i} $$ that has to be more compact than $$ ...
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votes
2answers
34 views

SU(2) is not isomorphic to $T^3$ [closed]

How can we prove that $SU(2)$ is not isomorphic to $\mathbb {S^1×S^1×S^1}$ by using the definition of $SU(2)$?
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0answers
15 views

G-invariant polynomials for a concrete G

The collection of matrices$$ \left( \begin{array}{lll} a^2 & 2ab &b^2 \\ ac & bc+ad & bd \\ c^2 & 2cd & d^2 \end{array} \right)$$ indexed by $a,b,c,d \in \mathbb{R}$ is a ...
5
votes
1answer
66 views

Lie groups where $x \mapsto x^2$ is a diffeomorphism?

In every Lie group $G$ the function $x \mapsto x^2$ is a local diffeomorphism in a neighbourhood of the identiy. (This is because its differential is: $v \mapsto 2v$ when considered as a map from ...
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0answers
17 views

Criterion for semi-simplicity of Lie algebra generated by vector fields

Suppose I have a finite collection of smooth vector fields $V:=\{V_1,...,V_k\}$ on a smooth manifold $M$. Moreover suppose that the Lie algebra $g$, generated by $V$ (where the Lie bracket is defined ...
1
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0answers
53 views

Lie Algebra SU(2)

Given a two dimensional Hilbert-space, $\mathcal{H}$, and a vector $\eta \in \mathcal{H}$, of this space, if $\eta$ transforms in SU(2) like this, $$\eta \rightarrow e^{(-i\alpha ...
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1answer
27 views

Cartan Lie Algebra of the Unitary Group $U(N)$?

I am having trouble understanding the Lie Algebra terminology. What is the Cartain Lie algebra of the unitary group $U(n)$? It must be in many textbooks, but they explain it very generally in terms ...
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2answers
51 views

Is Cartan subalgebra of Complex semisimple Lie algebra the maximal Abelian subalgebra? I found two places give the different answers.

In wiki https://en.wikipedia.org/wiki/Cartan_subalgebra Example 4, it says that Cartan subalgebra of complex semisimple Lie algebra is not maximal Abelian subalgebra. However in Brian C. Hall's ...
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1answer
18 views

Cohomology of complex Lie groups via compact form

Let $G$ be a compact Lie group. Let $G_{\mathbb{C}}$ be a complex Lie group such that there is inclusion $i: G \rightarrow G_{\mathbb{C}}$ of Lie groups. Moreover I require that differential of $i$ ...
3
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0answers
25 views

Minimal word length of factorization of invertible matrices into elementary matrices

Let $K$ be a field. As is well known, one can decompose every matrix $A \in GL(n,K)$ into a product of elementary matrices. By an elementary matrix, I mean a matrix which belongs to one of the ...
1
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1answer
16 views

dimension of space of origin-symmetric ellipsoids

I wonder how I can compute the dimension of the space of all origin-symmetric ellipsoids? What is the best approach?
0
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1answer
54 views

What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma}$?

In quantum mechanics we learn about the Pauli spin matrices: $$ \sigma_1 = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( \begin{array}{cc} i ...
0
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0answers
12 views

Calculation of commutator of Lie algebra for affine linear maps

This problem was answered before, but I'm stack with a technical point. Let $G$ be the Lie group of linear polynomials under composition (that is, affine transformations), $$\{x \mapsto ax+b, a\neq 0, ...
2
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0answers
26 views

Taylor series identity for polynomial using Lie group

The following question is from Kirillov's Introduction to Lie Groups and Lie Algebras, and my attempt is the following: $$\sum_{n\geq ...
2
votes
1answer
29 views

Non-equivalent metrics on $PSL_2(\mathbb{R})$

I am reading a paper on continued fractions and it uses the following result on Lie Groups: Fix an arbitrary left-invariant metric $d$ on $PSL_2(\mathbb{R})$ ... This phrase really throws me ...
2
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0answers
38 views

Why are $\mathfrak{pgl}_n\simeq\mathfrak{sl}_n$ when characteristic does not divide $n$?

Suppose $k$ is some algebraically closed field whose characteristic does not divide $n$. Why can we identify the lie algebras $\mathfrak{pgl}_n\simeq\mathfrak{sl_n}$ of the projective linear group and ...
2
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0answers
47 views

Standard Basis of $SU(2)$--where does the 1/2 come from?

The most common matrix representation of $SU(2)$ is given by $$ \begin{pmatrix} a & b\\ b^* & -a^*\\ \end{pmatrix} $$ where $a,b\in\mathbb{C}$. If we denote real components by the subscript ...
1
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1answer
48 views

Representation theory of Lie groups and outer automorphisms

If $G$ is a simply connected Lie group (I have in mind $G=SL_n(\mathbb{C})$), then we have an isomorphism $Aut(G)/Inn(G)\rightarrow Aut(g)/Ad(G)$ induced by taking the differential at $1$; here $g$ is ...
0
votes
0answers
16 views

Connected components of Lie group stabilizers

I'm trying to show that for $G$ a compact Lie group and $\alpha$ a transitive action of $G$ on a (Hausdorff) connected smooth manifold $X$, if there is $x\in X$ such that the stabilizer $G_x$ is ...
3
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1answer
70 views

Computational Topology and Lie Group Theory [closed]

I study Machine Learning and my limited background in math is enough to understand all the popular algorithms and methods. However, recently, Topology has been successfully applied to Data Analysis ...
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0answers
17 views

left and right jacobians (Not derivatives) of a Lie group

Let $G<Gl(n)$ be a Lie group, and let $g:\mathbb{R}^n\rightarrow G$ be a smooth curve, parametrized as $g(q(t))$ with $t\in\mathbb{R}$. I understand that in that case it holds that $$ \tag 1 ...
0
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1answer
29 views

Rotations of sphere $\mathbb S^2$

In the picture bellow; How to prove that the result of rotation about $P$ through angle $\theta$, followed by rotation about $Q$ through angle $\varphi$ is rotation about $R$ through some angle? ــ ...
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1answer
30 views

Examples of non linear Lie-Groups

When looking for a non linear Lie Group I always find the example of the Heisenberg Group $H$ modulo a normal Group $N$. Where the matrix of the two groups are of this form $$ H = \begin{bmatrix} 1 ...
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1answer
24 views

Logarithm of complex matrix

For invertible matrix $A$, we have $\log(\det A) = \mathrm{tr}(\log A)$ due to a corollary of Jacobi's formula. What if we had the argument $iA$ instead? Would the above relation still hold? Edit: ...
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votes
1answer
44 views

Is a set defined by equations always closed?

I've heard during a lecture that a "set defined by equations is always closed". The equation was a matricial equation of the type $$AA^T=I$$ The lecturer didn't spend more than this sentence on the ...
0
votes
0answers
50 views

Lie algebra of a finite group

I'm trying to find the normalizer of the Pauli group $G_n$ (as a subgroup of $SU(2^n)$) utilizing Lie algebras, as is done in a reference to find the normalizer of the Heisenberg group $HW(n)$. There, ...
3
votes
1answer
78 views

When is a lattice dense in a torus?

Let $\pi: \mathbb{R}^n\rightarrow \mathbb{R}^n/\mathbb{Z}^n$. What (necessary and sufficient) criteria on $A\in GL_n(\mathbb{R})$ guarantee $\pi(A\mathbb{Z}^n)$ is dense?
0
votes
1answer
16 views

Linear algebraic group inside $GL_n$

Let $G$ be a linear algebraic group. Consider the closed embedding $G \hookrightarrow GL_n$. Let $K$ be any field. Let $x \in GL_n(K)$. Now suppose we know that $x^n \in G(K) $ for some positive ...
5
votes
1answer
68 views

Smooth manifold which is a group, but not a Lie Group

Are there (preferably non-pathological) examples of smooth manifolds, which are groups, but not Lie groups? In books one can see plenty of examples of Lie groups, but I haven't seen an example where ...
1
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0answers
11 views

$A_\mathfrak{q}(\lambda)$ module and Zuckerman's derived functor module

Does any reference book give the explicit definition for $A_\mathfrak{q}(\lambda)$ module? Or are $A_\mathfrak{q}(\lambda)$ and Zuckerman's derived functor module the same thing?
6
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0answers
81 views

A subspace is invariant by the Lie group if it is invariant by the Lie algebra

Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie ...
1
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0answers
27 views

Commutator of Lie sub-algebra

I have a problem understanding the proof of Proposition 8.20, page 211, in Besse's Einstein Manifolds. He considers a semi-simple Lie algebra $\mathfrak{g}$ with $\operatorname{Ad}$-invariant scalar ...
2
votes
1answer
29 views

Push forward of the Lebesgue measure is the Haar measure of the Carnot group

I have the following problem. I have a Carnot group $(\mathbb G,*)$ which is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ is stratified as $\mathfrak g= ...
2
votes
1answer
64 views

How to show $\operatorname{Out}(\operatorname{SL}_3(\mathbb{C})) \cong \mathbb{Z}/2\mathbb{Z}$?

Let $G$ be a Lie group, $\operatorname{Aut}(G)$ the group of diffeomorphisms of $G$ that are also homomorphisms. Denote by $\operatorname{Inn}(G) \unlhd \operatorname{Aut}(G)$ the group of ...
0
votes
0answers
18 views

Isomorphism between semi-orthogonal group $O(2,2)$ and direct product of projective general linear group $PGL(2,\mathbb{R})$ with itself

Is there any natural isomorphism between semi-orthogonal group $O(2,2)$ and direct product of projective general linear group $PGL(2,\mathbb{R})=GL(2,\mathbb{R})/R^+$ with itself?, where $O(2,2)$ is ...
2
votes
0answers
28 views

Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
0
votes
0answers
10 views

Arrive to the group law in exponential coordinates using the vector fields expressed in exponential coordinates

I need an help with the following question. I have this definition for Engels group $\mathbb E$: it is the only connected and simply connected Lie group that has the Engels algebra $\mathfrak g$ as ...
0
votes
1answer
56 views

Constructive proof that the Lie functor is faithful?

I am wondering how to show that the Lie functor taking Lie groups to Lie algebras is faithful? In particular, I am looking for a constructive proof, since I am working in the context of synthetic ...
1
vote
1answer
29 views

Applications of $SO(3)$ irreps to spatial rotation

I've been on a kick learning about Lie Groups, with special emphasis on $SO(3)$ recently. I work in the field of spacecraft attitude determination and control, where is $SO(3)$ of interest in the ...
1
vote
0answers
41 views

Basis for traceless, symmetric matrices?

Consider, for example, the set of of all symmetric, traceless $4 \times 4$ matrices. I'm trying to find a correctly normalized basis for this set. So far, I have $$s(1)=\left( \begin{array}{cccc} 0 ...
1
vote
0answers
18 views

Where can I find free text to learn Lie group analysis for solving nonlinear systems of differential equations?

Can anyone please recommend to me free online text for learning Lie group analysis for solving nonlinear systems of differential equations? it is preferable that text is not too complicated, it would ...
0
votes
0answers
22 views

Differential of right action map on a manifold?

Let $P$ be a smooth manifold, $G$ a Lie group and $$\mu:P\times G\longrightarrow P, p\longmapsto \mu(p, g):=p\cdot g,$$ a right action of $G$ on $P$.Furthermore, suppose $\alpha:I\longrightarrow P$, ...
1
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0answers
15 views

Homogeneous Space structures of the sphere

I'm reading through these lecture notes and trying to apply the following Theorem: A Lie group $G$ acts globally and transitively on a manifold $M$ if and only if $M \cong G/H$ is isomorphic to the ...
2
votes
0answers
30 views

Is $S_3$ an exceptional Lie group?

Up until now I have had the belief that finite groups do not supply meaningful examples of Lie groups. However in this paper, Kostant claims that the symmetric group on three elements is an ...
0
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1answer
28 views

Treatments of Lie theory and Noether theorems in tensor notation

I am looking for a conceptual treatment of Lie theory and Noether theorems that uses tensors calculus rather than exterior calculus. I know that tensor calculus is not optimal for these subjects, ...
9
votes
2answers
210 views

Kähler metrics on the coadjoint orbits of a compact Lie group

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. It is well-known that each orbit for the coadjoint representation of $G$ on $\mathfrak{g}^*$ carries a canonical symplectic structure, ...
1
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0answers
42 views

The condition for the exponential map of Lie group is surjective or injective

$G$ is a connected Lie group, $g$ is its Lie algebra. 1) What is the necessary and sufficient condition for the exponential map from $g$ to $G$ is surjective? 2) What is the necessary and ...
0
votes
1answer
14 views

Normal subgroup of Engel group

The Engel algebra $\mathfrak g$ is the Lie algebra generated, as a vector space, by four vectors $X_1,X_2,X_3,X_4$ with the only non trivial commutation relations:$$[X_1,X_2]=X_3, \quad ...
0
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0answers
42 views

Advantages and disadvantages of defining Lie bracket via right invariant vs. left invariant vector fields

I was just wondering what are the advantages and disadvantages of the two conventions used for defining Lie brackets? For example, if we use right invariant vector fields as the convention for ...