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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81 views

Does triality survive in product Lie groups?

Look at the following diagrams of Lie groups ...
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3answers
238 views

Lie algebra of $GL_n(\mathbb{C})$

I would like to do Tao's exercise 6 (i) but before I can even attempt it I need to be clear about his terminology. Exercise 6 Show that the Lie algebra $gl_n(\mathbb{C})$ of the general linear group ...
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2answers
395 views

“Change of basis” from standard vector space to matrix Lie algebra, and its inverse

For matrix Lie groups, the exponential map is usually defined as a mapping from the set of $n \times n$ matrices onto itself. However, sometimes it is useful to have a minimal parametrization of our ...
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1answer
207 views

Lie Groups which are not Hausdorff

I suspect this isn't a terribly difficult question, but I don't know the answer and I'd guess someone has already looked into it. Is it possible for a Lie group on a non-Hausdorff manifold to exist? ...
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6answers
3k views

Jacobian matrix of the Rodrigues' formula (exponential map)

I am working an algorithm which is supposed to align a pair of images. The motion model, which describes the pose $p$ of an image (with respect to the second) in 3D space, is purely rotational. ...
3
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1answer
450 views

Derivative of function including matrix logarithm

Is the following equation a first order approximation or incorrect for general matrix Lie groups? And what are the higher order terms? $$\frac{\partial}{\partial\mathbf x} (\log(\mathtt ...
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1answer
592 views

De Rham Cohomology of a Lie group

If $G$ is a connected Lie group with Lie algebra $\mathcal{G}$, then de Rham cohomology of left invariant différential forms $H_L^*(G)$ is isomorphic to the Chevalley–Eilenberg cohomology ...
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1answer
196 views

Lie group structure on some topological spaces

I have some basic background in Lie theory and I have some difficulties to show that some topological spaces admits a Lie group structure. More precisely, for a given Lie group $G$: 1) Why its ...
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1answer
526 views

First and second homotopy groups of a connected Lie group

I try to understand why for a connected Lie group $G$ the first fundamental group $\pi_1(G)$ is abelian, and mainly why the second fundamental group is trivial $\pi_2(G)=0$? Thanks for anyone who ...
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174 views

Why connected Lie groups are homotopy equivalent to connected compact Lie groups?

I am looking for a simple proof of a Mostow Theroem, which asserts that any connected Lie group $G$ admits a maximal compact subgroup $K$ (which is necessarily connected) such that $$G\simeq ...
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1answer
211 views

Lie group of matrices commuting with a fixed set of matrices

Given a set of $S$ of $ n\times n$ Hermitian matrices with entries in $\mathbb{C}$, the set of all $n\times n$ unitary matrices that commute with $S$ forms a Lie group $G$. My question is what Lie ...
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1answer
152 views

Relation between a Lie group and Lie algebra representation for $W \otimes V$

We can define a representation of a Lie group and get the induced representation of the Lie algebra. Let $G$ act on $V$ and $W$, $\mathfrak{g}$ be the Lie algebra associated to $G$ and $X \in ...
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1answer
2k views

Description of SU(1, 1)

For a homework question, I am required to "describe the Lie group SU(1, 1)". This is a bit ambiguous, but I think what that means is I need to find a parametrisation of the elements of the group. I ...
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1answer
206 views

Group of units of a Clifford algebra

Let $V$ be a finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, $Q$ a quadratic form on $V$ and $\mathrm{Cl}(V,Q)$ the corresponding Clifford algebra. Now consider the multiplicative ...
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321 views

Elementary proof of the third Lie theorem

I would like to understand a (not well known) proof due to G.M; Tuynman, of the third Lie theorem, which asserts that for any given finite dimensional Lie algebra $\mathcal{G}$ there exists a (simply ...
4
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2answers
170 views

Compact connected Lie groups have no continuous square roots

Let $G$ be a group. We say that $f : G \to G$ is a square root of $G$, if $f(x)^2=x$ for all $x \in G.$ Prove that a compact connected Lie group $G$ has no continuous square root. What if, ...
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2answers
261 views

Reference request: group theory

Currently I'm studying differential geometry and PDEs - so I often meet the use of groups. I also studied symmetries methods for solutions of differential equations but the connection between Lie ...
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1answer
350 views

How to obtain a Lie group from a Lie algebra

Please pardon me, if the question is too simple. How can we obtain a Lie group form a given Lie algebra (say in 3 dimension) in practise? Can someone illustrate it in 3 dimension? Is there a ...
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1answer
222 views

Representation which have no unique decomposition into irreducible

What kind of examples of groups and representations should I keep in mind, which do not uniquely decompose into irreducible representations? I am mostly interested in characteristic zero ...
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1answer
151 views

Schwartz kernel theorem for induced representation/ Schur algebra for locally compact groups

Given a finite group $G$ and subgroups $H$ and $K$, and representation $$\sigma: H \rightarrow GL(V_\sigma), \qquad \pi: K \rightarrow GL(V_\pi).$$ Now consider the space of functions $f: G ...
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0answers
146 views

Describe Geometrically the action of a discrete subgroup on the complex Heisenberg Group

I am new to the study of lie groups, nilmanifolds etc. but the following question can be described very basically I think. Take a complex Heisenberg group $G$ $$ \left( \begin{array}{ccc} 1 & ...
5
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0answers
260 views

Short exact sequences of topological groups and Lie groups

could someone please clarify the definitions of extensions of topological groups and Lie groups. For topological groups, what I see in most papers is as follows: An extension of topological groups $0 ...
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1answer
1k views

Unitary representations of non-compact Lie groups

This question is somewhat of a continuation of this question that I had asked earlier - Representations of a non-compact group are labeled by its maximal compact subgroup? I want to know when or is ...
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1answer
713 views

The mathematics behind Clebsch-Gordan Coefficients

In quantum physics we have to work a lot with Clebsch-Gordan coefficients and generalizations like the Wigner 3j,6j, and 9j symbols. In our coursework we are taught that the coefficients are coupling ...
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2answers
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Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
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141 views

Does the algebraic structure of a Lie group restrict the possible dimensions of other Lie groups isomorphic to it?

In a recent question, I initially doubted that $\mathbb{C}^\times\cong S^1$, my intuition being that $\mathbb{C}^\times$ has one more "dimension" than $S^1$ - in rigorous terms, $S^1$ is (or rather, ...
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1answer
468 views

Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
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1answer
175 views

Finding double coset representatives in finite groups of Lie type

Is there a standard algorithm for finding the double coset representatives of $H_1$ \ $G/H_2$, where the groups are finite of Lie type? Specifically, I need to compute the representatives when ...
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2answers
369 views

On “complexifying” vector spaces

I think this question should be quite trivial. For some reason I did not really get the author's argument. I shall use the symbols from the book to avoid ambiguity. In the book "Lectures on Lie ...
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1answer
195 views

Non-algebraic Lie groups

When trying to learn about Lie groups I find that most natural examples of Lie groups are actually examples of algebraic groups. What are some interesting examples of Lie groups which are not ...
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145 views

Lie group quotient bundle with image of normalizer as structure group

In Glen Bredon "Topology and Geometry", Ch. II-13, I am stuck on the following "Problem 1. Finish the proof at the end of this section that $G\rightarrow G/H$ is a bundle. Also show that this is a ...
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2answers
193 views

lie groups and topology

Is there a relationship between Lie groups and topology and is there a succinct explanation that can be provided? Is there a good online reference that discusses this.
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2answers
375 views

Why is the identity component of a matrix group a subgroup?

I'm working through Stillwell's "Naive Lie Theory". I'm supposed to show that the identity component of a matrix group is a subgroup in two steps. I'm allowed to assume that "matrix multiplication ...
6
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1answer
148 views

What is the proper way to address this result?

Reading a paper I came through an argument proving the following: Let be given a smooth action of $\mathbb{R}^n$ on a manifold $M$, such that it is infinitesimally free and its orbits are ...
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1answer
314 views

Proof that Lie group with finite centre is compact if and only if its Killing form is negative definite

I am gathering material for an exposition on Lie theory and I am after proofs that a Lie group with finite centre is compact if and only if its Killing form is negative definite. I know of one, ...
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1answer
90 views

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)?
3
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1answer
435 views

Levi-Civita connection of a left-invariant metric

How do I compute Levi-Civita connection of a left-invariant metric on a Lie group in a neighbourhood of $1$ by knowing only its Lie algebra and the metric form on it? I know it's possible because a ...
5
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391 views

Alternating and special orthogonal groups which are simple

I have an idle curiosity about a funny coincidence between two sequences of groups. It is well-known by those who know it well that the alternating group $A_d$ of degree $d$ is simple if and only if ...
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2answers
192 views

Concerning the definition of effective quotient orbifold

I've been trying to figure out orbifolds, and in all of the sources I seem to be confused with the orbifold structure on quotient orbifolds. A quotient orbifold is defined as follows. Let $M$ be a ...
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0answers
81 views

$X_G$ is a CW complex

I know the following result: If $X$ is a compact smooth manifold and $G$ is a compact Lie group which acts smoothly on $X$, then $X_G = (X\times EG)/G$ is a CW complex. I don't know how to ...
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1answer
298 views

the volume of 3-sphere

My question is why the volume of 3-sphere that lies between a rotation angle of $\theta$ and $\theta+d\theta$ is $$2\pi \sin^2(\theta/2) d\theta$$
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2answers
164 views

Atlas and transition maps for a Lie group

Consider the group $G$ consisting of real $2\times 2$ matrices of the form $\begin{pmatrix} x & 0\\ y & z \end{pmatrix}$ of nonzero determinant with multiplication as the operation. I ...
7
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1answer
266 views

Can any of the exotic differentiable structures on $\mathbb R^4$ make $GL(\mathbb R^2)$ into an 'exotic' Lie group structure?

I am just beginning to learn about Lie groups and am made somewhat uncomfortable by the textbook's handwavy decision to talk about Lie groups $GL(V)$ where $V$ is some $n$-dimensional real vector ...
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1answer
334 views

Closed form for the exponential of a Lie algebra 3x3 matrix?

Matrices of the form: $$\begin{pmatrix} iz+iy&-iz+w&-iy-w\\ -iz-w&iz+ix&-ix+w\\ -iy+w&-ix-w&iy+ix\end{pmatrix}$$ where $x,y,z,w$ may be assumed to be real, form a Lie ...
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0answers
250 views

How to show that the structure constant of SU(3) is invariant?

So suppose $f_{ijk}$ is the antisymmetric structure constant of SU(3), and $D^8_{ij}(g)$ is the matrices of 8-dimensional adjoint representation of SU(3), then how to show that ...
0
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1answer
449 views

What is meant by adjoint of a linear transformation w.r.t a given inner product?

Consider a matrix Lie group equipped with a left &/or right invariant metric. The adjoint of linear transformation $A$ with respect to the inner product is denoted as $A^*$. Here what is ...
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3answers
863 views

Lie algebra of a quotient of Lie groups

Suppose I have a Lie group $G$ and a closed normal subgroup $H$, both connected. Then I can form the quotient $G/H$, which is again a Lie group. On the other hand, the derivative of the embedding ...
4
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1answer
99 views

Continuous transformations of a triangle bound on $S_1$

Suppose we take the circle $S_1$ and three points on this circle, which defines a triangle. By moving the points continuously on $S_1$, we obtain a continuous transformation of the triangle. I was ...
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3answers
492 views

Does every irreducible representation of a compact group occur in tensor products of a faithful representation (and its dual)?

Let $G$ be a compact (Hausdorff) group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. (Hence $G$ is a Lie group.) Is it true that every irreducible representation ...
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164 views

Are Lie Groups Homogeneous Spaces?

Is any Lie Group a homogeneous space?