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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Elements in finite symmetry groups share a common fixed point

I am reading Tapp's intro to matrix groups for undergraduates. On page 46 he states the following theorem: For $X\subseteq \mathbb R^2$ if $Symm(X)$ is finite then it is isomorphic to $D_m$ or ...
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21 views

Question about Modular function in Haar measure

I'm reading the book "Basic Lie Theory" (http://guests.mpim-bonn.mpg.de/abbaspou/Lie-Book_verrouille.pdf) and I'm trying to understand the proof of Lemma 2.3.4 which states that: Let $G$ be a locally ...
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19 views

How is the periodic structure of SO(n) reflected to its lie algebra so(n)?

An element of $SO(n)$ represents an rotation so that it must have identity with $2\pi$-like additional rotation. On the other hand, the elements of lie algebra $so(n)$ construct an noncompact vector ...
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10 views

Direct symmetries of $X\subseteq \mathbb R^n$

I don't quite understand the definition of direct symmetries for a subset $X$ of $\mathbb R^n$. In this book(page 45) the definition is given as follows: $$ Symm^+ (X) := \left \{ ...
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2answers
33 views

Lie Group Automorphism which are diffeomorphism

Is every smooth automorphism of a Lie Group $G$ a diffeomorphism?
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11 views

Jacobian calculation for function over SE(3)

For an Extended Kalman Filter implementation I need to calculate the Jacobian of $$ f(C) = CC_1 $$ where $C, C_1 \in SE(3)$. In [1] is explained that that the following equation holds: $$ ...
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38 views
+200

How many permutations do we need before we're in $SU\left( n\right)$?

Let $\mathcal{L}\subseteq \mathfrak{su}\left( n\right)$ be a Lie algebra for $n \geq 2$ with Lie group $G = e^{\mathcal L}$, and let $X \in G$ be represented by an $n\times n$ matrix (I prefer fixing ...
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14 views

Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
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21 views

Find the lie derivative of a particular integral

Let $\mathbb{X}=(1,y)$ be a vector field on $\mathbb{R}^2$. Let $\Phi_t$ be the flow of $\mathbb{X}$. The flow of $\mathbb{X}$ I have calculated to be $\Phi_t(x,y)=(x+t,ye^t)$ Given a function ...
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27 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
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exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...
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42 views

Why is there no normal, dense, totally disconnected subgroup of $SO(n)$?

There are two exercises in Stillwell's Naive Lie Theory that I'm having trouble doing: 3.8.4: Show that the subgroup $H = \{ \cos 2\pi r + i\sin 2\pi r : r\text{ rational} \}$ of the circle $SO(2)$ ...
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0answers
41 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
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1answer
25 views

How to define the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$?

I read a paper and on page 9, the paragraph before Proposition 5, it is said that let the vector field $\hat{X}$ on a manifold $M$ defined by an element $X$ in a Lie algebra $\mathfrak{g}$. How to ...
4
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0answers
75 views

One parameter subgroup that leaves every compact set is a proper map

If a one parameter subgroup $\phi:\mathbb{R}\rightarrow G$ of a Lie group $G$ comes back infinitely often to a compact set $K$, is it all contained in a compact set? I think $\phi(\mathbb{R})K\subset ...
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13 views

Finding an overgroup or a subgroup in PGL

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$. Let $G=PGL_4(k)$. Let $H=\{ \small\left[\begin{array}{cccc} x & ...
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1answer
16 views

The Weyl group and eigenspaces

Let $V$ be a representation of the Weyl group. For any reflection $\sigma_{\alpha}$ (where $\alpha$ is a root), we know that $V$ has two eigenspaces with eigenvalues $1$ and $-1$. The ...
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48 views

Lie group quotient structure

Let $G$ be a Lie group and $H$ a normal finite subgroup. Let $\pi : G \to G/H$ be the quotient surjection. How would one show that $G/H$ is a Lie group?
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15 views

Normal subgroup invariant under $\text{Ad}_g$

Denote by $G$ a Lie group with corresponding Lie algebra $\text{Lie}(G)$. There the three maps inner automorphism/conjugation: $\text{Int}_g = L_{g^{-1}} \circ R_g \in \text{Aut}(G)$, $\text{Ad}_g ...
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1answer
32 views

The action of a Lie algebra on a manifold is a Lie algebra homomorphism. How to show it?

By definition, the action of a Lie algebra $\mathfrak g$ on a manifold $M$ is a Lie algebra homomorphism, $\mathcal A: \mathfrak g\rightarrow\mathfrak X(M), \xi\mapsto\xi_M$ such that the action map ...
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27 views
+50

Finding $J$ such that this diagram commutes

I did this exercise here and need someone to check if my work is correct: Is it possible to find a matrix $J\in M_{2n}(\mathbb C)$ such that the following diagram commutes? $$\begin{array} \mathbb ...
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0answers
21 views

Useful Coordinate Families on Lie Groups

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. We all know, since $\exp$ is a diffeomorphism in some neighborhood $V$ of $0\in\mathfrak{g}$, that we can cover $G$ in coordinate charts ...
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One more question about mapping quaternionic matrices into real matrices

Real matrices that lie in the image of the inclusion homomorphism $\rho_n: M_n(\mathbb C) \to M_{2n}(\mathbb R)$ are called complex linear. It is easy to see that a real matrix is complex linear if ...
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2answers
28 views

On the inclusion homomorphism for quaternionic matrices into complex matrices

My thoughts / background information: It is easy to find an inclusion homomorphism for complex matrices into real matrices: considering the one dimensional case note that multiplying a complex number ...
1
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1answer
25 views

Is the center of a compact Lie algebra precisely the set of vectors on which the Killing form is zero?

Suppose a Lie algebra $\frak{g}$ has a killing form, $B$, which is negative semidefinite. Suppose $B(X,X)=0$ for some $X\in \frak{g}$. Is $X$ necessarily in the center of $\frak{g}$?
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1answer
26 views

How does one define weights for a semisimple Lie group?

For compact Lie groups one considers a maximal torus to define the weight space decomposition of a representation. For a complex semisimple Lie algebra one considers a Cartan subalgebra. How does ...
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35 views

Norm preserving matrices also preserve inner products

I am trying to prove that if $A \in M_n(\mathbb C)$ preserves norms then it also preserves inner products. I showed this for real matrices and I want to use this for this proof here. Let $f_n: \mathbb ...
1
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0answers
20 views

Reference request: classification of simple Lie groups and simple real Lie algebras

I am trying to understand the classification of simple Lie groups and the theory of highest weights for semisimple Lie groups by first understanding the case for complex Lie algebras, then relating to ...
4
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0answers
38 views

Convolution of matrix coefficients is also a matrix coefficients

I have a question about the convolution of matrix coefficients as follows: Let $G$ be a compact Lie group. A Map $f:G\rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite ...
2
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34 views

Intuition behind complex inner product

Let $f_n : \mathbb C^n \to \mathbb R^{2n}$ be defined by $$(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$$ I am having trouble believing that $$ \langle X, Y\rangle_{\mathbb ...
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1answer
34 views

Group law, cubics and Lie group

Let $C$ be a smooth complex cubic in $CP^{2}$. We know that there is a group structure by using the intersection of projective lines (cf. Ried, Undergraduate AG, Section 2), which is really different ...
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1answer
12 views

On the definition of quaternionic-linear real matrices

I'm reading Tapp's introduction to matrix groups. The book introduced complex-linear matrices. Let me reproduce the definition in my own words: Let $B\in M_{2n}(\mathbb R)$. Let $J$ be the matrix $$ ...
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0answers
14 views

Proof for necessary and sufficient condition that $B\in M_{2n}(\mathbb R)$ is complex linear

Let $f_n : \mathbb C^n \to \mathbb R^{2n}$ be defined by $$(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$$ I showed that $f_n$ is a $\mathbb C$-linear isomorphism. I want to ...
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0answers
59 views

Proof that this diagram commutes

This is sort of a follow up on another question of mine. Let me give the full details: Define an $\mathbb R$-linear isomorphism $f_n : \mathbb C^n \to \mathbb R^{2n}$ by $$(x_1 + iy_1, x_2 + iy_2, ...
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0answers
18 views

What do you get when you pull the Bruhat Decomposition back to the Lie algebra via the exponential map?

If $G$ is a connected, reductive, complex group with Borel subgroup $B < G$ and Weyl group $W$, we can write $$G = \bigsqcup_{w \in W} B w B$$ If $\mathfrak{g}$ is the Lie algebra of $G$, we have ...
1
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1answer
31 views

Isomorphic Lie algebras have isomorphic centers

I think that if two Lie algebras are isomorphic, then their centers should be isomorphic - is this true? I am sure the answer is obvious to those in the know! Here is my attempt at a proof which looks ...
1
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1answer
18 views

Is $\mathfrak{o}(n)$ a subalgebra of $\mathfrak{u}(n)$?

A quick simple question to start the weekend (I hope). The Lie algebra $\mathfrak{u}(n)$ is the set of $n\times n$ skew-Hermitian matrices over $\mathbb{C}$ and the Lie algebra $\mathfrak{o}(n)$ is ...
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0answers
38 views

Lie Bracket of vector fields on Lie group

Let $H$ be a Lie group and $\mathfrak{h}$ its Lie algebra. Given a smooth function $v: H \to \mathfrak{h}$, define the vector field $\bar{v} : H \to TH$, $h \mapsto d(R_{h})_{e} v(h)$, where $R_{h} : ...
3
votes
1answer
71 views

Relation between Aut(G) and Aut(g)

Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. We know that when $G$ is simply connected, $\mathrm{Aut}(G)=\mathrm{Aut}(\mathfrak{g})$ (this should follow from the fact that we can ...
0
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1answer
48 views

Does the matrix exponential take open sets into open sets?

This is from Hall's Lie Groups, Lie Algebras, and Representations, in theorem $2.13$: Let $B_\varepsilon$ be the open ball of radius $\varepsilon$ about zero in $M_n (\mathbb{C})$ [$= ...
2
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1answer
15 views

The derived algebra is a Lie subalgebra

A (hopefully) very simple question that has been bugging me all day! Let $L$ be a Lie algebra then the derived Lie algebra $L'$ is $$ L' = \{ \, [u,v] : \forall u,v\in L \, \}. $$ I want to show ...
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25 views

How to construct explicit matrix representations of $\mathfrak{su}(3)$

I'd like to implement an algorithm which produces matrix representations of the (complexified) Lie Algebra $\mathfrak{su}(3)$ on carrier spaces with arbitrary highest weight vector; i.e. 8 $n\times n$ ...
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1answer
50 views

Distance of subgroup to element in Lie groups

Given a (compact, closed) Lie group $G$ and a (closed) subgroup $H$, what is the distance of the identity to $Hg$ (or $gH$), where $g\in G$ and $Hg$ denotes the orbit under left-multiplication? The ...
1
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1answer
20 views

Prove that the tangent space to group of unipotent matrices is a subspace of M(2,R).

Given the set of unipotent matrices: $S = \left\{ A\in GL_{2}(\mathbb{R}) \;:\; A=\left( \begin{matrix} 1 & a \\ 0 & 1 \end{matrix} ...
3
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0answers
39 views

Is a connected unipotent subgroup always contained in a Borel subgroup?

As the question says, is a connected unipotent subgroup $U$ of a linear algebraic group scheme $G$ always contained in a Borel subgroup of $G$? I have an argument for why the answer is yes, and I ...
3
votes
2answers
58 views

Action on sheaf cohomology in Bott-Borel-Weil theorem

Let $G$ be simply connected complex semisimple Lie grou and $P \subseteq G$ parabolic subgroup. Suppose $V$ is finite dimensional irreducible representation of $P$ with highest weight $\lambda$, and ...
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1answer
44 views

More on rotation matrices: a basis for $SO(3)$?

Consider the group $SO(3)$. The rotation around the $x$-axis is represented by the matrix $$R_x = \left ( \begin{array}{ ccc } 1 & 0 & 0 \\ 0 & \cos \Theta & - \sin \Theta \\ 0 & ...
2
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1answer
20 views

Weyl Groups/Borel

Could someone tell me where to find a proof of the following statement that I found in some notes about characteristic classes I was reading? If $G$ is a compact connected Lie group with maximal ...
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0answers
15 views

A compact group with a finite dimensional faithful representation [duplicate]

Theorem: If $G$ a compact group has a finite dimensional faithful representation $W$, then any irreducible representation $V$ is contained in $W(k,l) = W^{\otimes k} \otimes (W^*)^{\otimes l}$ for ...
4
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0answers
49 views

A very difficult problem about the existence of following $SU(2)$ matrices?

Let $G_i$ be a sequence of $2\times2$ $SU(2)$ matrices, where $i=1,2,...,n$; and $P$ represents a permutation of $\left \{ 1,2,...,n \right \}$. The question is: Does there exist a sequence of ...