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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25 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
2
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1answer
32 views

Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Let $G$ ge a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogenuous space $\frac{G}{H}$ are connected, then $G$ is connected. Remark: For this proof I use the following ...
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2answers
66 views

Calculating the differential of the inverse of matrix exp?

Let $A(t)$ and $B(t)$ be two matrix-valued smooth function satisfying the equation, $B(t) = e^{A(t)}$. I need to express $\frac{dA(t)}{dt}$ in terms of $B(t)$. I know that there is a formula of ...
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1answer
31 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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1answer
32 views

Why does $(X_a,Y_b)(f\circ \mu(a,b)) = X_a(f\circ R_b(a)) + Y_b(f\circ L_a(b))$?

On the first page of these notes: if $\mu\colon G\times G\to G$ is the multiplication map on a Lie group $G$, then given a point $(a,b)\in G\times G$ and letting $R_b$ and $L_a$ denote ...
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1answer
23 views

Proving that Lie groups are locally connected

I'm trying to show that if $G$ is a Lie group, then it is locally connected, i.e., for any point $p$ in an open subset $U$ of $G$, there is a connected neighborhood $V$ of $p$ such that $V\subset U$. ...
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27 views

References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
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1answer
30 views

Is there a harmonic analysis theory for the group of strictly increasing continuous invertibe functions from $\mathbb{R} \mapsto \mathbb{R}$

I have very little to add to the title, would appreciate pointers to the literature. A follow up question I have is: If one considers differentiable strictly increasing functions, do they lend ...
4
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1answer
38 views

Differentiability of an action of the group of invertible elements of a $C^{*}$-algebra $\mathcal{A}$ on the dual of $\mathcal{A}$

I am studying the actions of Banach-Lie groups on Banach manifolds, and I am not able to concretely evaluate the differentiability properties of a specific action. Let $\mathcal{A}$ be a unital ...
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0answers
40 views

Why can we find a basis for the elements of the Lie algebra?

I am a physicist and we do Lie algebras pretty informally, so I hope my question makes any sense to a mathematician. There is one thing that I don't quite understand, which is why we can find a basis ...
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0answers
20 views

How to construct free subgroup of SO(3) using Baire's Theorem?

I'm looking to demonstrate that there exists a free group on two generators $ G\subseteq SO(3) $ (this is for homework, so hints are preferred to complete answers I've turned it in now, so full ...
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1answer
48 views

Commutation relation between su(N) and clifford algebra generators.

Why does the $ \gamma_5 $ matrix commute with the generators of the $su(N)$ algebra? In the case of the chiral symmetry from physics, [$Q_a$, $Q_b^5$] = $i \epsilon_{abc}Q^5_c$ where the $Q_a, Q^5_a$ ...
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0answers
38 views

Compact Lie group $G$ with Lie algebra $\frak g$ satisfying $gZg^{-1}=-Z$ for $Z\in\frak g$ and $g\in G$

Let $G$ be a compact Lie group with Lie algebra $\frak g$. Are there known conditions on $G$ guaranteeing the following property: $$ \hbox{For each $Z\in\frak g$ there exists an element $g\in G$ ...
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1answer
15 views

The Lie algebra of the generalized unitary group $\{g \in GL_n(\mathbb{C}) : gS\bar{g}^t=S\}$ is $\{XS+S\overline{X}{}^t=0\}$

Let $ S \in M_n(\mathbb{C}) $ be a square matrix and let $ X$ be in the Lie algebra $\mathbb{\mu(S)} $ of the generalized unitary group, $$U(S):=\{g \in GL_n(\mathbb{C}); gS\bar{g}^t=S\} .$$ ...
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1answer
28 views

Adjoint matrix in $\mathbb{so_3}$

$\mathbb{so_3}$ has the following basis: $X_1=\begin{bmatrix} 0 & & \\ & &1 \\ & -1 & \end{bmatrix}$, s: $X_2=\begin{bmatrix} & & 1\\ & 0& \\ -1 & & ...
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1answer
32 views

Show that $ (\pi(g)\phi)(v)=\phi({^t}gv) $ defines a representation

Let $ G=SL_2(\mathbb{C}) $ and consider the action of $ G $ on the space of smooth functions on column vectors $ v \in \mathbb{C^2} $ given by: $ (\pi(g)\phi)(v)=\phi({^t}gv) $ Question 1: Show that ...
1
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1answer
34 views

Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis

For a lie algebra $\mathbb{g} $ we can define the adjoint representation as: $ ad: \mathbb{g} \rightarrow End(\mathbb{g}) $ as the map such that $ad_x(y)=[x, y] $ for all $\in \mathbb{g} $ I am ...
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0answers
22 views

Lie algebra of $SL_2(\mathbb{R})$ and show $\exp(X)=I+X$ where $I \in SL_2(\mathbb{R}) $ and $X \in sl_2(\mathbb{R})$

I am doing an undergraduate course on Representation Theory and am trying to solve these consecutive questions. The first two I am ok with (I just included them for context), but I could do with some ...
3
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0answers
23 views

Isometries of the canonical left invariant metric on $GL_n$

Consider $GL_n(\mathbb{R})$ with the left-invariant Riemannian metric obtained by left translating the standard metric on $T_IGL_n \cong M_n \cong \mathbb{R}^{n^2}$. (i.e for $X,Y \in T_IGL_n \,: ...
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1answer
30 views

G2 as algebra of endomorphisms preserving a trilinear form

I am trying to find some literature or papers about the topic in the title. I've read multiple times that the Lie-Algebra G2 can be described in such a way, but I've yet to find some good, ...
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30 views

Existence of a canonical map of quadratic forms

For $X=\mathbb C^N\oplus \mathbb C^N$ equipped with a real structure $J^2=1$ and symplectic structure $S$ satisfying $$J(z_1,z_2)=(\bar z_2,\bar z_1),~~~~~S(z_1,z_2)=(z_1,-z_2)$$ we see that $X$ has ...
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2answers
63 views

Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $

I read in a paper that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $ this is counterintuitive / surprising since $\mathrm{SO}_3(\mathbb{R}) \not \simeq \mathrm{SL}_2(\mathbb{R}) $ ...
2
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1answer
42 views

Lie Group and a basis of tangent space

Let $G \subset M(3\times 3, \mathbb{R})$ be the space of all matrices of the form $\left( \begin{array} &1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1\end{array}\right) $ where ...
1
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1answer
27 views

The Lie Algebra of Invertible Upper Triangular Matrices

From Wikipedia: The Lie algebra of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a solvable Lie algebra. I ...
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0answers
46 views

Formula for differential of $\exp$ at a Banach algebra.

In Rossman (Lie Groups - An introduction through linear groups), he makes the following statement: Theorem: $$\exp'_X(Y)=\exp(X)\frac{1-\exp(-ad_X)}{ad_X} Y,$$ where ...
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0answers
36 views

Constructing the Bruhat-Tits building of $SL_2$

I am trying to understand how to construct the Bruhat-Tits building of $SL_2(\mathbb{Q}_p)$. I am reading the general construction of the semisimple Bruhat-Tits building of a connected reductive ...
2
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37 views

Is there an example of a non compact, semisimple, amenable Lie group?

By semisimple I mean the real Lie algebra of $G$ is semisimple. I guess there is not but I can't formulate a rigorous argument.
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1answer
36 views

A Lie group associated to a matrix via semi direct product

Assume that $A \in M_{n}(\mathbb{R}) $ is a matrix. Then $A$ generates a one parameter (with parameter $t\in \mathbb{R}$) family of group automorphisms of $\mathbb{R}^{n}$ with $x\mapsto ...
0
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1answer
43 views

When does an affine manifold inherit a (quotient) group action?

By an affine manifold I mean a real $n$-dimensional manifold $M$ with charts whose transition functions are in the affine group $Aff(\Bbb R^n)$. There are several other equivalent definitions ...
1
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1answer
31 views

Is $SU(1) = U(1)$ or the trivial group?

The $S$ denotes usually determinant $1$ and therefore I'm not sure how one defines it in the one-dimensional case. Is $SU(1)$ actually the same as $U(1)$, because it makes no sense to talk about ...
0
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0answers
39 views

Closure of algebraic groups

Let $\phi: G\rightarrow V$ an embedding, with $G$ a complex algebraic group and $V$ a vector space (actually a $G$-representation). Is it true that the closure (in the Zariski topology) of $\phi(G)$ ...
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0answers
10 views

derivation of basic jacobian in SO(3) at omega=0

Every time I go a website on on Lie Group SO(3), it is mentioned that $$ \left.\frac{\partial}{\partial \mathbf \omega}\exp(\hat {\mathbf \omega})\right|_{\mathbf \omega=\mathbf 0} = ...
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0answers
30 views

simply connected abelian Lie groups

Is there any simple proof of the fact that: Up to isomorphism $\mathbb R^n$ is the only simply-connected abelian Lie group? By simple proof I mean using only basic concepts from groups and topology.
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1answer
47 views

Why does $\frac{d}{dt}e^{X+tY} |_{t=0}$ depend linearly on $Y$ with $X$ fixed?

I'm studying the proof of Baker-Campbell-Hausdorff formula from Brian Hall's book Lie Groups, Lie Algebras and Representations. I am stuck at this part: I don't get why continuity of exp implies ...
3
votes
3answers
53 views

Direct calculation of the tangent space of $SO(3)$

Let $SO(3)$={$RR^T=I$, $det(R)=1$}, I need to show that a base of the tangent space in the identity is given by: $$E_i=\frac{d}{dt}\exp(tL_i)|_{t=0}$$ where $$L_1= \left(\begin{matrix} 0& 1& ...
3
votes
1answer
41 views

finding high weight vector in Verma module

Let $\frak{g}$ be a (semi-)simple lie algebra. Let $\lambda$ be a dominant integral weight. Denote $L(\lambda)$ to be the irreducible representation of highest weight $\lambda$. From BGG resolution, ...
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2answers
38 views

Existence of Ad-invariant bilinear form gives a certain Lie algebra homomorphism

Let $G_1 \subset G$ be Lie groups and $\mathfrak{g}_1, \ \mathfrak{g}$ the corresponding Lie algebras. Assume that there is a non-degenerate bilinear form $\langle \cdot, \cdot \rangle$ on ...
2
votes
1answer
53 views

If an orbit $G\cdot x$ is closed in the standard topology, is it Zariski-closed?

Let $G$ be a complex linear algebraic group acting linearly on finite-dimensional complex vector space $V$. If an orbit $G\cdot x$ is closed in the standard topology on $V$, is it also ...
2
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1answer
75 views

The quotient of $\Bbb R^3$ by a finite group.

Let $\Gamma$ be a finite subgroup of $SO(3)$ acting on $\Bbb R^3$. What sort of space do we get by taking the quotient $\Bbb R^3/\Gamma$? Is that a manifold? The group $\Gamma$ is compact since it is ...
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0answers
35 views

Dual basis cotangent space

I have been given the unitary sphere in the Euclidean space. $$F(\theta, \phi) =(\sin\theta \cos\phi, \sin\theta \sin\phi,\cos\theta)$$ I'm asked to show that the dual base of $E_1=F_*(\partial ...
3
votes
2answers
58 views

Smooth Manifolds and Lie Group Action

I have started to read about Lie group action on smooth manifolds. A question popped up in my mind and I am not sure it's silly or not. Diff (M) = space of all smooth diffeomorphisms is a large ...
1
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1answer
37 views

Manifolds with $GL_n(\mathbb{R})$-action.

What is the condition on $n$-dimensional real manifolds in order that they admit an $GL_n(\mathbb{R})$-action in the sense of https://en.wikipedia.org/wiki/Lie_group_action that resembles the ...
2
votes
1answer
45 views

What is the topology on $O(3,1)$?

I've read that there are four connected components of $O(3,1)$. Then, if I'm not mistaken, if $A \in O(3,1)$, then to determine which connected component $A$ is in we look at whether $\text{det}(A)= ...
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0answers
27 views

Quotients by simply connected closed subgroups

I have come across an exercise asking for a proof of something that is definitely false: If $G$ is a Lie group, $H$ a connected closed subgroup and $G/H$ simply connected, then $G$ is itself simply ...
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1answer
22 views

Left action on $G/H$ is proper?

Let $G$ be a Lie group and $H\subseteq G$ a closed subgroup. Then, $G/H$ is a smooth manifold and inherits a smooth action of $G$: $$G\times G/H\longrightarrow G/H,\quad g_1\cdot(g_2H)=g_1g_2H.$$ ...
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0answers
37 views

Invariance of a Vector Field under the action of a Group

I've got a one-parameter group given by \begin{equation} \theta_{t}\left(x,y,z\right)=\left(e^{t}x,e^{t}y,e^{t}z\right) \end{equation} I already have th infinitesimal generator vector field ...
2
votes
0answers
67 views

Weyl group of complex Lie group

Let $G$ be a compact connected Lie group with maximal torus $T$. The Weyl group is defined by $$W:=N_G(T)/T.$$ Now, $G$ has a complexification $G_{\Bbb C}$ with maximal torus $T_{\Bbb C}$ which is the ...
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0answers
12 views

Two nilmanifolds of the same Lie group

By a nilmanifold I mean a quotient $M =\Gamma \backslash G$ of a connected, simply-connected nilpotent real Lie group G by the left action of a maximal lattice 􀀀, i.e. a discrete cocompact subgroup. ...
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0answers
31 views

$SO(3)$ and twisting the 2-sphere

I am currently reading some parts of "Rotating Relativistic Stars" by Friedman and Stergioulas and I have to say mathematics should NOT be taught by astrophysicists... Anyway, I've encountered the ...
5
votes
0answers
48 views

Find the extended form of the group generated by an operator?

I tried to find the extended form of the group generated by the following operators. (I): The first operator $$A=z\frac{\partial }{\partial z}+1$$ To find the extended form of the group ...