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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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
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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)= ...
1
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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 ...
1
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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
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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 ...
1
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0answers
38 views

Adjoint orbits are described by minimal polynomials

I've read that if $\frak g$ is a complex semisimple Lie algebra and $G$ a Lie group with Lie algebra $\frak g$, and $H\in{\frak g}$ is regular semisimple, then the adjoint orbit $${\cal O}_H=\{{\rm ...
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0answers
25 views

Automorphism of semisimple Lie algebra corresponding to a simple reflection

Let $\mathfrak{g}$ be a complex, finite-dimensional Lie algebra. Let $\mathfrak{h}\subset \mathfrak{g}$, $W$ and $\Pi$ be a Cartan subalgebra, its Weyl group and the set of all simple roots, ...
4
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1answer
81 views

Is the special orthogonal group really rationally homotopy commutative?

It is a classical result that the rational cohomology of $SO(n)$ is given by: $$H^*(SO(2m); \mathbb{Q}) = \begin{cases} S(\beta_1, \dots, \beta_{m-1}, \alpha_{2m-1}) & n = 2m \\ S(\beta_1, \dots, ...
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0answers
24 views

Show $\phi$ is a isomorphism as a lie algebra homomorphism

Show $\phi$ is a isomorphism as a lie algebra homomorphism $\phi: \textbf{su}_2 \bigotimes_{\mathbb{R}} \mathbb{C}\rightarrow sl_2(\mathbb{C})$ and $\phi: a(I \bigotimes 1)+b(J \bigotimes 1)+c(K ...
0
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1answer
25 views

Root system independent of chosen Cartan algebra

I have read on "Lectures on Lie groups and Lie algebras" (by Carter, Segal, Macdonald) that Cartan subalgebras are related by some automorphism of the Lie algebra and this is proved using a density ...
1
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0answers
33 views

Symmetric Polynomials in Geometry

I'm interested in symmetric polynomials. Could you name some nice examples in Differential Geometry where they are clearly useful? I would also be interested in algebraic examples if connected with ...
0
votes
1answer
20 views

Borel density theorem

I know the following version of Borel density theorem: If $G$ is a connected real Lie group such that every continuous homomorphism from $G$ to a compact group is trivial, and if $H$ is a closed ...
0
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0answers
29 views

How many subalgebras are there in $sl_3$?

The Lie algebra $sl_3$ is 8 dimensional and $B=\{h_1, h_2, e_1, e_2, [e_1, e_2], f_1, f_2, [f_1, f_2]\}$ is a basis of $sl_3$. For every $x \in B$, $\text{Span}\{x\}$ is a one-dimensional subalgebra ...
0
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1answer
85 views

Isomorphism between homotopy groups of Lie group, Grassmann manifold

It is asserted without proof in a book edited by Novikov and Rokhlin that $$\pi_{k - 1}(\text{GL}_n^+(\mathbb{R})) \cong \pi_k(\tilde{G}_n).$$ I know how to show that these two spaces are bijective. ...
0
votes
0answers
50 views

Do we have $\{g x g^{-1}: x \in \mathfrak{g}\} = \mathfrak{g}$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix $g \in G$. Do we have $\{gxg^{-1}: x \in \mathfrak{g}\} = \mathfrak{g}$? Thank you very much. Edit: I think that the answer is yes. We ...
1
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1answer
50 views

Strange parametrization of SO(3)

I have this parametrization of the sphere that seems quite a mess \begin{equation} R_{ij}=\cos r\left(\delta_{ij}-\hat{\theta}_{i}\hat{\theta}_{j}\right)+\hat{\theta}_{i}\hat{\theta}_{j}+\sin ...
5
votes
0answers
70 views

Pauli Matrices as Representations of Reflection Operators?

How does one derive, using, say, the Householder transformation $$ R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + ...
0
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0answers
15 views

Lie group actions

I am looking for a nice reference to study the action of a Lie group $G$ on a smooth manifold $M$, $\psi : G\to\mathrm{Diff}(M)$: Linearization: in a neighborhood of a fixed point, what we can ...
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0answers
27 views

When a group of ismorphisms is a Lie group

What are the known cases where a group of isomorphisms of a smooth manifolds (diffeomorphisms that respect a given structure on the manifold) is a Lie group? such as: isometries of a compact ...
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0answers
23 views

Differential of a Lie group homomorphism

If $f : G\to H$ is a Lie group homomorphism, what can we say about its differential $d_ef : \mathcal{G}\to\mathcal{H}$? Is it a Lie algebra morphism or anti-morphism? ...
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1answer
23 views

Dimension of the indefinite orthogonal group [closed]

How to find the dimension of the group $O_{p,q}(\mathbb R)= \{g \in GL_n(\mathbb R): g^TI_{p,q} \ g = I_{p,q}\}$, where $I_{p,q}= diag(1,..., 1,-1,...,-1)$ and $p+q=n$?
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0answers
23 views

Lie group of differential operators

I have the following three partial differential operators $$A=y \frac{\partial}{\partial y}$$ $$B=y^{-1}(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial y}+c-1)$$ ...
1
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0answers
14 views

Basis vectors for “perturbed slicings” of a function, using SE(3)

Given a function $\Phi: \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\mathbb{R}^2\rightarrow\mathbb{R}$, using a rigid mapping $s \in SE(3)$ such that for ...
1
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0answers
14 views

evaluationmap is a submersion iff lie group acts transitively on connected components

Let $M$ be a manifold and $G$ a connected Liegroup acting smoothly on $M$. Take $x \in M$ and define by $ev_x \colon G \to M, \ g \mapsto g.x$ the evaluationmap at $x$. Is it true, that $G$ acts ...
0
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1answer
21 views

Unitary dual of $\{0\}$ and $\mathbb R$

How to prove that, the unitary dual of $\{0\}$ and $\mathbb R$ are the trivial identity representation $id$ and the representation $\chi_x (y) = e^{i x y}; y\in \mathbb R$, respectively. Thank you ...
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votes
2answers
65 views

prove $RP^3\cong SO(3)$ [closed]

Suppose $RP^3$ is the real 3-dimensional projective space,prove the rotation group $SO(3)$ is homeophoric to $RP^3$.
1
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1answer
37 views

prove $GL(n,R)\cong O(n)\times R_{+}^n\times R^{\frac{n(n-1)}{2}}$

$GL(n,R)$ is the general linear group ,$O(n)$ is the orthogonal group,how to prove $GL(n,R)\cong O(n)\times R_{+}^n\times R^{\frac{n(n-1)}{2}}$
2
votes
1answer
75 views

How to compute the center of $SU(2)$?

It is stated in our lecture notes without proof that the center of $SU(2)=\{\pm 1\}$. I understand how to find the center of $SO(3)$, which is $\{1\}$ and that is given in the notes, is that somehow ...
5
votes
1answer
53 views

The category of Lie algebra representations

A representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$ is a homomorphism of Lie algebras $\mathfrak{g} \to \mathfrak{gl}(V)$. We define morphisms between representations as ...
0
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0answers
24 views

Jet-groupois, left translation and smoothness of a vector field

In this question I work in the smooth category. For basic definitions on lie groupids see "General Theory of Lie Groupoids and Lie Algebroids" by Kirill Mackenzie. Let us consider the Lie groupoid $G ...
6
votes
0answers
139 views

Duistermaat & Kolk's lost chapters on Lie groups

I am reading the book Lie Groups by Duistermaat and Kolk. It is written in the preface that "the text contains references to chapters belonging to a future volume". I could not find this second ...
3
votes
0answers
48 views

Two group structures on the cotangent bundle of a Lie group. Are they related?

Let $G$ be a compact connected Lie group, and consider its cotangent bundle $T^*G$. There are two ways of viewing this space. Using left translation, we can trivialize $T^*G\cong G\times{\frak ...
0
votes
1answer
49 views

Matrix representation of Heisenberg group

Equiped with the law $(a,b,c)\circ (a',b',c') = (a + a', b + b', c + c' + ab'),$ the matrix representation of the Heisenberg group $H^3$ is given by $$ \begin{pmatrix} 1 & a & c\\ 0 & ...
0
votes
0answers
25 views

Lie group action and Lie algebra action.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $r \in g \otimes g$ and $b \in g$. Consider the adjoint action $g \times g \otimes g \to g \otimes g $ given by $(b, x, y) \mapsto b.(x \otimes y) = ...
3
votes
2answers
67 views

Why do we require that a simple Lie algebra be non-abelian?

We say that a Lie $k$-algebra is simple if it is a simple object in the category of Lie algebras, and also nonabelian. The only simple object which we do not consider to be a simple Lie algebra under ...
0
votes
1answer
39 views

What is the manifold underlying the Lie group $SU(p,q)$?

I've been trying to google around this topic without success, apologies in advance if I missed an obvious resource. I'm trying to understand what manifold (compact or not) underlies the complex Lie ...
0
votes
0answers
35 views

How does this product of matrices define a local diffeomorphism?

Let $$H_1 := \left\{\begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix} \mid x \in \mathbb{R}\right\},$$ $$H_2 := \left\{\begin{pmatrix} 1 & 0 \\ y & 1\end{pmatrix} \mid y \in ...
0
votes
1answer
26 views

Chevalley basis for $G_2$

I want to find the Chevalley basis for the exceptional group $G_2$. Could you point to literature where the computation is done in detail or show me how to do it?
1
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2answers
37 views

What can we say about $Aut(G)$ for an arbitrary Lie group $G$?

Let $G$ be a Lie group, $\mathfrak g$ be its Lie algebra, and $Aut(G)$ be the group of its smooth automorphism. Then, my questions are: (1) Is $Aut(G)$ again a smooth manifold? and particularly a Lie ...
3
votes
1answer
27 views

What the expression of a one-dimensional representation of $H$

Let $G= \{ g=(x,y,t); \quad x,y,t \in \mathbb R\}$ be the Heisenberg group and $H= \{ g=(x,y,t) \in G; \quad x=0\}= \{ h=(0,y,t); \quad y,t \in \mathbb R\}$ be a subgroup of $G$. I want to know why ...
1
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1answer
45 views

A proof that every compact Lie Group has torsion second homotopy group

I am trying to prove that every compact lie group has torsion second homotopy group but I get stuck. My argument is the following: Since $\Pi_1(G)$ is finite the universal cover G* of G is also a ...
1
vote
1answer
33 views

Are surjective Lie group homomorphisms which induce isomorphisms of Lie algebras covering maps?

Question: Suppose that $\varphi:G\to H$ is a surjective Lie group homomorphism whose differential $\varphi_*:{\frak g}\to{\frak h}$ is a Lie algebra isomorphism. Is $\varphi$ necessarily a smooth ...
0
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0answers
19 views

Splitting of short exact sequence. with the existence of non-deg. bilinearform

Let $\mathfrak{h}$ be a subalgebra of $\mathfrak{g}$,and $j \colon \mathfrak{h} \to \mathfrak{g}$ be the injective Liealgebra-Homomorphismus. Assuming now, that we have a non-degenerate bilinearform ...
2
votes
0answers
47 views

embeddings $SU(2) \to SU(N)$ and representations

How can we prove that group immersions $SU(2) \to SU(N)$ (up to conjugacy) are in 1-1 correspondence with (non-trivial) $N$-dimensional representations of $SU(2)$ (up to equivalence)? Feel free to ...
0
votes
1answer
22 views

Complexification of maximal compact subgroup of $GL(2,\mathbb{R})$.

Given the Lie group $G=GL(2,\mathbb{R})$, we have that $K=O(2)$ is a maximal compact subgroup of $G$. I am trying to describe the complexification $K_\mathbb{C}$ of $K$. The Lie algebra $k_0$ of $K$ ...
1
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0answers
19 views

Composition of analytic functions is analytic in Manifolds

My problem is in analytic manifolds.According to Cohn's book a function $f$ in a manifold $M$ is analytic at $p \in M$ if it can be expressed as a power series of $\sigma(p)=(x_{0})$. That means ...