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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how to find an integral curve in Lie group?

Given a Lie group $G$, $e$ is its identity element and $g$ is one element of $G$. I want do find a curve $\gamma(t)$ that satisfies these conditions: 1) passes $g$ and $e$, that is ...
2
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0answers
19 views

The product of Weyl groups is the Weyl group of the product

Let $G$ and $H$ be compact, connected Lie groups. Write $W(-)$ for the Weyl group. Then it is not hard to see that $$W(G \times H) \cong W(G) \times W(H).$$ Where can I find a published statement of ...
2
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1answer
60 views

Tangent space and differential forms of quotient groups

I have difficulty understanding the following argument because it seems that many details are swept under the rug and I am looking for a detailed rigorous exposition on this (I'll try to make clear ...
2
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1answer
69 views

how to extend a vector at $e$ of a Lie group to a left invariant vector field?

I am reading some books about Lie group and Lie algebra. Denote the set of all the left invariant vector fields as $\mathfrak{X}_L$, and the tangent space at $e$ of $G$ as $T_eG$. They say that the ...
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34 views

Chevalley group

Let $L=\mathfrak sl_6$ be the special linear Lie algebra over $\mathbb C$ and let $S=\{\alpha_1, \alpha_2, \alpha_3,\alpha_4,\alpha_5\}$ be a set of simple roots. Then the set of positive roots are ...
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46 views

Fundamental representation of $O(3)$

I want to check if the fundamental representation of $O(3)$ is irreducible on $\mathbb{R}^3$ and $\mathbb{C}^3$. I want to use isomorphism properties. I know this relation exists $$ ...
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55 views

representatives of $\pi_n(U(n))$

I know that $\pi_n(U(n)) \cong \mathbb{Z}$ for $n$ odd. I am looking for a description of an explicit map $S^n \rightarrow U(n)$ that represents the generator in this group. By precomposing with maps ...
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1answer
26 views

Tangent space of closed subgroup of Lie group upon action

I am trying to show the below statement which I very strongly feel should be "obviously correct", but I think I am missing the easy way to see this. Let $G$ be a Lie gorup and $H$ a closed subgroup. ...
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51 views

Geodesics on Homogeneous Spaces

Consider a homogeneous space $G/\text{Stab}_p \cong M$ where $G$ is a compact Lie group active transitively on $M$ (a compact manifold). If $F$ is a Finsler Metric on $G$ which pushes forward ...
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22 views

Reference on the Crystallographic restriction theorem and some related results

Firstly I precise that I am working on $\mathbb C$ the plan of complex numbers I have some result for which I look for references (mathematical books or articles) where the reader can find their ...
2
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1answer
37 views

Lie Subgroup Example - Explanation?

I'm currently working through Jeff Lee's 'Manifolds and Differential Geometry'. He defines a Lie Subgroup, $H$, to be an abstract subgroup of a Lie Group $G$, such that the inclusion map ...
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125 views

Reconstructing Lie group globally from the exponential map

This should be an elementary question in Lie group theory, although I'm pretty sure I'm mixing up concepts. Any help clarifying would be much appreciated. Set up Let $G$ be finite-dimensional real ...
3
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0answers
24 views

Iwasawa module: $O[[G]]$ is noetherian where $G$ is a compact $p$ adic Lie group

Let $G$ be a compact $p$ adic Lie group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Then how to show that $O[[G]]$ is noetherian. I was reading the article here and on ...
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18 views

Embedding of classical Lie groups

This is somehow very natural question so I'm sure that the answer should be well known: Whitney theorem states that each (say paracompact) $n$-dimensional manifold could be embedded in ...
1
vote
1answer
63 views

how to calculate a vector in a left invariant vector field?

I would like to understand the left invariant vector field by using a numerical example. Now we consider a Lie group $G=SE(3)$, and the associated Lie algebra is $\mathfrak{g}=se(3)$. We suppose: ...
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0answers
26 views

injectivity of the map covering the inclusion $SO(n)\subset GL^+(n)$

Let $n\ge 2$ and $\theta\colon Spin(n)\rightarrow SO(n,\mathbb{R})$ be the two-fold covering of $SO(n,\mathbb{R})$ by the spin group $Spin(n)$, $\tilde{\theta}\colon ...
0
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1answer
12 views

Decreasing sequence in subgroup

Can someone clarify the following statement: Let $H\subset\mathbb{R}$ be a subgroup $\neq 0$. Let $a = \text{inf} \lbrace x\in H \vert x>0 \rbrace$. If $a\notin H$ then there exists a decreasing ...
2
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1answer
51 views

Lie groups and Lie algebras

Ok so I'm confused about the relation between these two concepts. If I have a Lie Group $G$ I can associate a Lie algebra $\mathfrak{g}$ by taking his tangent space in the identity, with the ...
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0answers
16 views

Finding coefficients for this Lie algebra isomorphism

This is a question closely related to my previous questions. How, in this thread here did Hee Kwon Lee find the coefficients $(-i,1,0),\ (-i,-1,0)$ and $(0,0,2i)$? In the linked thread ...
4
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2answers
59 views

How to use the Killing form to write down a Lie group isomorphism, and what is the induced Lie algebra isomorphism?

This is a follow up question on my previous question here. One of the answers suggests that I find a map $$SL_2(\mathbb{C}) \to SO_3(\mathbb{C})$$ and then the map induced by this map will be a Lie ...
1
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1answer
27 views

Solving a large system of linear equations to satisfy the Lie bracket: am I doing it right?

I'm still working on a Lie algebra isomorphism from the Lie algebra of $SL_2(\mathbb C)$ into the Lie algebra of $O(3, \mathbb C)$. It has been suggested to me to use linear combinations of ...
3
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1answer
55 views

Lie algebra homomorphism: is my understanding correct?

Using answer to my previous question I made some progress towards understanding Lie algebra homomorphisms. But of course I am unsure whether my thoughts are really correct so again I'd like to request ...
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1answer
34 views

Doubt in Peter Olver “Applications of Lie groups to differential equations”

Book: Applications of Lie groups to differential equations. Second edition (1993). Page: 117-120. Chapter: 2. Section 2.4: Calculation of symmetry groups. Example: 2.41. The heat equation. Question ...
2
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1answer
20 views

Lie subalgebras of $so(3)$ and complex Lie subalgebras of $sl(2,\mathbb{C})$.

I am looking for a reference that: describes all Lie subalgebras of $so(3)$,and describes all complex Lie subalgebras of $sl(2,\mathbb{C})$. Does someone have appropriate references?
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1answer
40 views

Zeroth homotopy group of the space $O(3)/H$

Short version of the question: Is $\pi_0(O(3)/C2) = Z_2$ as $\pi_0(O(3)) = Z_2$? Here $C_2$ is a cyclic group of order 2. Long version or the question: The zeroth homotopy group describes the ...
0
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0answers
37 views

positive roots remain positive

Let $W$ be the Weyl group of $SL_{n+1}$ and $w \in W$. Let $R^+$ denote the set of positive roots with respect to the Borel subgroup of upper triangular matrices. Define $R^+(w)=\{\alpha \in R^+: ...
2
votes
1answer
47 views

Is the exponential map ever not injective?

Let $G\subseteq GL_n(\mathbb R)$ and let $\mathfrak g$ denote its Lie algebra. Let $e: \mathfrak g \to G$ be the map $X \mapsto e^X$. Does there exist an example of $G$ and $\mathfrak g$ such ...
2
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0answers
49 views

Smooth manifolds having an analytic structure.

I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are ...
3
votes
2answers
189 views

Defining an isomorphism that respects the Lie bracket: is my work correct?

I previously determined that if $\mathfrak{sl}$ denotes the Lie algebra of $SL_2(\mathbb C)$ and $\mathfrak o$ denotes the Lie algebra of $O(3,\mathbb C)$ then a basis for $\mathfrak{sl}$ is given by ...
0
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0answers
17 views

Unimodular Lie group property based on the self-adjoint application

Let $\{e_1, e_2, e_3\}$ a pseudo-orthonormal basis of $\mathcal g$, definined the linear transformation $L:\mathcal g \rightarrow \mathcal g$ such that $L(e_1)=[e_2,e_3]$, $L(e_2)=[e_3,e_1]$, ...
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0answers
21 views

Determining whether a Lie algebra is also a complex Lie algebra

I am trying to learn Lie theory. In the following I will share my thoughts. Please, can you check my work for correctness and point out any mistakes to me? I am trying to determine whether ...
0
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2answers
22 views

Center of compact lie group closed?

Let me specify that my knowledge about Lie groups/algebras is reduced to bits and pieces I learned from various diff geometry textbooks. I could not find a reference for the following question (I am ...
2
votes
1answer
41 views

Is a Lie algebra a complex or a real vector space?

I am trying to learn Lie theory and for this purpose I worked out the Lie algebras of some matrix groups. The examples I worked happened to be complex matrix groups and it lead me to wonder whether, ...
2
votes
0answers
32 views

Determining Lie algebra: are my thoughts correct?

Let $UT_1(n, \mathbb R)$ denote the set of all upper triangular real matrices with diagonal equal to $1$. This is a Lie group. I am trying to determine its Lie algebra. Please can you tell me if ...
0
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1answer
29 views

Lie algebra of the semiorthogonal group $O(p,q)$ [closed]

How do I prove this: If $\mathcal{O}(p,q)$ is a Lie algebra of the semiorthogonal group $O(p,q)$ then $\mathcal O(p,q)$ consist of all matrices of the form: $$X= \left( \begin{matrix} a ...
2
votes
0answers
23 views

Do I have the right idea for this isomorphism of Lie algebras of matrix groups?

I previously determined that the Lie algebra of $O(3,\mathbb C)$ is the set of skew symmetric matrices and that the Lie algebra of $SL_2(\mathbb C)$ is the set of traceless matrices. I am now trying ...
2
votes
1answer
39 views

Where is my mistake: determining the Lie algebra of complex orthogonal matrices

I tried to determine the Lie algebra of $O(3, \mathbb C)$ but I think there is a mistake but I can't find it. Here is my work: Let $\mathfrak o$ denote the Lie algebra of $O(3, \mathbb C)$. The ...
1
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1answer
78 views

Example of a linear algebraic group which is not a Lie group

I am trying to reconcile the notions of algebraic groups, linear algebraic groups, Lie groups, and Lie algebras, along with their notions of root systems, maximal tori, etc. To begin, I am trying to ...
0
votes
1answer
43 views

Proof that these two definitions are equivalent

Where can I find a proof of or how can I prove that these two definitions are equivalent? Definition 1: The Lie algebra of a Lie group $G \subset GL_n$ is the tangent space at $I$. Definition 2: ...
1
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0answers
33 views

Can someone intuitively describe the fiber bundle and product spaces of SO(3)?

I have zero understanding of differential geometry or topology so the material found online are useless for me. So in light of that can someone use very general terms or analogy to comment about the ...
1
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1answer
51 views

Are complex numbers a trivial lie group of itself? [closed]

Let $z$ be a complex number, then let's define a map $e^{T(*)}$. Let $w = e^{T(z)}$, where $T$ is some real number. Then is $z$ a lie group of $w$?
0
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1answer
53 views

Lie Groups: Differential Operations

Given a Lie group. Multiplication and inversion act infinitesimally at the identity by: $$\mathrm{d}\mu:\mathrm{T}_{(e,e)}(G\times G)\to\mathrm{T}_eG:(u,v)\mapsto u+v$$ ...
0
votes
1answer
31 views

Injective acton of an Chevalley generator of $\mathfrak{sl}(2)$ on non-intergral weight module

I have a problem as follows. Let $E_{i,j}\in M_2(\mathbb{C})$ be the elementary matrix and $\mathfrak{g}:=\mathfrak{sl}(2) = \{ A \in M_2({\mathbb{C}})| tr(A) =0 \}$ the special linear Lie algebra. ...
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2answers
53 views

How to “find” this Lie algebra: proof that $\mathfrak{sl}$ is trace zero matrices

I saw this table here on Wikipedia and it states that the Lie algebra of the special linear group $SL_n(\mathbb C)$ is the group of traceless matrices $\mathfrak{sl}_n$. I know the definition of a ...
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0answers
21 views

Is there always a g in a compact connected Lie group whose powers equidistributes in G?

I'm starting to understand some basics things about equidistributed sequences and i found my self asking this question on the basis of the example of the torus and Weyl equidistribution theorem: ...
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1answer
52 views

Restriction functors between Categories O over semi-simple Lie algebras

I have the following question: Let $\mathfrak{g}:= \mathfrak{gl}(3)$ be the general linear algebra and $\mathfrak{gl}(2) \cong \mathfrak{a} \subset \mathfrak{gl}(3)$ a sub-algebra. Let ...
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0answers
20 views

How does the $10$ dimensional irrep (tensor) of $SU(3)$ look like?

We know that for $SU(3)$ the following tensors furnish the $\mathbf{d}$ dimensional irreducible representation: $$\phi^i\hspace{1cm} (\mathbf{3})\\ \phi^{ij}\hspace{1cm} (\text{asymmetric in ...
1
vote
1answer
65 views

Typo on Wikipedia? Dimension of $U(n)$

Let $U(n)$ denote the unitary group. That is, $$ U(n) = \{A \in GL_n(\mathbb C)\mid A^\ast A = I\}$$ Wikipedia states: "The unitary group $U(n)$ is a real Lie group of dimension $n^2$. " There ...
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21 views

A question about finite simple groups of Lie type

I have a question about finite simple groups of Lie type. By the classification theorem of finite simple groups, we know that there are four important class of finite simple groups as follows: ...
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2answers
30 views

Basic sanity check: dimension of Lie groups / tangent spaces

A potential typo in an exercise prompted me to question my knowledge of manifolds. So what I need is a sanity check. Here is what I used to think before I got unsure: If $M$ is an $n$-manifold then ...