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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The isomorphisms between $S^5$ and $SU(3)/SU(2)$?

What is the precise isomorphisms between the coset $SU(3)/SU(2)$ and the five-sphere $S^5$?
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How to prove that the adjoint group is a Lie subgroup of $Gl(\mathfrak{g})$

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $Ad: G \rightarrow GL(\mathfrak{g})$ be the Adjoint representation. I want to prove that $Ad(G)$ is a Lie subgroup of $GL(\mathfrak{g})$. ...
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37 views

Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
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Tangent space to lie group at identity.

I'm supposed to show that for a Lie group G, $T_{(e,e)}G\times G \simeq T_eG\oplus T_eG$ and that $T_{(e,e)}m$ is given by $(X,Y)\mapsto X+Y$. I'm having trouble proving this. I'm not exactly clear ...
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8 views

Straight forward derivation of the bch formula?

Im doing a project on rigid body dynamics and need to derive the bch formula, anyone know a simple yet complete derivation?
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19 views

Adjoint representation of the isotropie group of a homogeneous space

I have difficulties seeing why is the following true: Let $G$ be a lie group and $H$ a closed subgroup, with $\tilde{g}$ and $\tilde{h}$ their lie algebras. The adjoint action of $g\in G$ is given by ...
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Gluing of two geodesic space along a proper space is geodesic.

Let $X_1$ and $X_2$ geodesic metric space glued along A a proper subspace of both and then given the pseudo metric. Why is the glued space geodesic? Any hint ? For notation and details one can see ...
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25 views

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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25 views

Exponentials of Representations of Lie Algebras

Assume G is a lie group and g is its lie algebra. Consider a representation of G : D:G->End(V). Then there is a corresponding representation of g : d:g->End(V). My question is, when you can express ...
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32 views

Compactness of Lie groups

Let $G$ be a Zariski-closed subgroup of $GL(V)$, where $V$ is an $n$-dimensional complex vector space. Question. Does $G$ have the structure of a compact Lie group? Such $G$ certainly is a Lie ...
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Are the orbits of a connected Lie group acting on a vector space always embedded manifolds?

Setting: We have a connected Lie group $G$ and a smooth map $G \to GL(V)$, where $V$ is a finite-dimensional vector space. Are the orbits of $G$ on $V$ embedded submanifolds? More precisely, if one ...
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Why is the space of symmetric positive definite matrices under the affine invariante metric a symmetric space?

I consider here the space of symmetric positive definite matrices SPD(n) with the metric invariant under: $(G,M) \rightarrow GMG^t $, where $M\in SPD(n)$ and $G\in Gl(n)$. The isotrpie group of $I$ is ...
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21 views

classifying map of associated bundles

Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$. Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$. Let $f_\xi: X\to BG$ be the ...
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1answer
27 views

Geodesic completeness of a Lie group

Let $G$ be a Lie group and $\rho$ some left(right, bi)-invariant Riemannian metric on $G$. Is it possible to say for which $\rho$ an underlying manifold $G$ is geodesically complete (maybe for every ...
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82 views

Differential in Lie groups

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...
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Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
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duality for (co)homology of Lie algebras

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, ...
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1answer
30 views

complexification of $SO(2)$

While computing the complexification of Lie group $SO(2)$, I get the result is all the matrix of the following form $$\left(\begin{array}{cc} \frac{e^{t-\sqrt{-1}\theta}+e^{-t+\sqrt{-1}\theta}}{2} ...
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1answer
42 views

Example of a proper metric space such that the associated length space is not proper.

Can anyone give me a example of a proper metric space $(X,d)$ such that $(X,\bar d)$, where $\bar d$ is the induced length metric, is not proper. I have a example but I am not sure if it is right. ...
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1answer
40 views

Correspondence between one-parameter subgroups of G and TeG

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
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1answer
30 views

Example of 3-dimensional nonabelian Lie group

Could you please give an example of a Lie group diffeomorphic to $S^1\times \mathbb{R}^2$? Okay, $S^1\times \mathbb{R}^2$ suits us. What about nonabelian one?
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Two definitions of left-invariant vector fields of a Lie group

I am reading these lines from a text which shows why the bracket of two left-invariant vector fields is also a left-invariant vector field. But cannot easily get one of the lines. Let $L_a$ be the ...
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21 views

Computations of common isometry groups, $O(n)/O(n-1), SO(n)/SO(n-1), U(n)/U(n-1)$, etc?

On wikipedia, some of the common isometry groups are given: $S^{n-1}\cong O(n)/O(n-1)$, $S^{n-1}\cong SO(n)/SO(n-1)$, etc. Is there a reference where some/any of these groups are computed? I'm just ...
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142 views

What does the logarithm of the identity look like on a Lie group?

Let $G$ be a compact, connected Lie group with identity element $e$ and $\mathfrak g$ its Lie algebra. Consider the set $$ L=\{A\in\mathfrak g\setminus\{0\};\exp(A)=e\}. $$ The most descriptive name ...
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If $df_e: T_eG_1\to T_eG_2$ is surjective, and $G_2$ connected Lie group, then $f$ is surjective.

Suppose $f$ is a morphism of Lie groups, and $df_e\colon T_eG_1\to T_eG_2$ is a surjective map of the tangent spaces of two Lie groups, where $G_2$ is connected. I read that by the Inverse Function ...
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1answer
75 views

A question about contractibility of a Lie group

In Lawson's book Spin Geometry, chapter II, before Proposition 1.11, it mentions a fact that if the 1st, 2nd and 3rd homotopy groups of a Lie group $G$ vanish (although $\pi_2(G)=0$ automatically), ...
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32 views

Why do Ad(K) orbits in the $-1$ eigenspace of a Cartan decomposition intersect the Weyl chamber once?

Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra ...
2
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1answer
55 views

When is “being a linear algebraic $k$-group” preserved?

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$. What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example When ...
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1answer
40 views

symmetric group as a subgroup of general linear group

Is the symmetric group $S_n$ a normal subgroup of the general linear group $GL(n,\mathbb{R})$? We regard $\sigma\in S_n$ acts on $\mathbb{R}^n$ by permuting the coordinates ...
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42 views

Identifying a Lie algebra from its universal enveloping algebra

Its been a while since I've worked on Lie algebras and I can't remember how to approach this problem: How do I identify the lie algebra (up to isomorphism) associated to a certain universal ...
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17 views

Component of the identity is generated by any connected neighborhood of the identity when the group is locally connected?

I read a theorem that if $G$ is a locally connected group, then the component of the identity $G_0$ is generated by any connected neighborhood of $e$. It goes like: Let $V$ be a connected nbhd of ...
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1answer
37 views

Rotations in higher dimensions vs. Spheres

Where my questions stem from: When we study the rotations in a plane or of some specific higher dimensions, there exists a neat approach to represent all the rotations as a spheres $\mathbb S^i$, for ...
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29 views

What Lie groups have a discrete set of order two elements?

We know that the set of order two elements of $R^n$, tori and $S^3$ are discrete. Are there others examples of Lie groups with such property? Are there some characterization of such class?
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Spaces of homogeneous type is seperable

Can anyone suggest me some reference for the following proof? $(X,d,\mu)$ is a space of homogenoeus type. Prove that it is seperable.
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Why is SO(3) simple

The question is just that. We define $SO(3)$ as the linear group of linear orthogonal maps, e.g. $\{ A\in GL(3, \mathbb{R}) | A A^T = Id, \, \det A = 1\}$ Why is this group a simple group, e.g. With ...
2
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1answer
23 views

Lie algebra and left-invariant vector fields

I want to prove that the tangent space of a Lie group at its identity $e$ is isomorphic to the vector space of left-invariant vector fields. Given an element $D \in T_e G$ (a derivation), the ...
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1answer
31 views

Reflection in terms of simple reflections

Suppose $\beta=\sum_{i=1}^ka_i\alpha_i$, where $\alpha_i$ are simple roots. Is there any easier way to write the reflection corresponding to $\beta$ say $s_{\beta}$ in terms of $s_{\alpha_i}$'s. I ...
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1answer
24 views

A smooth non-stably trivial smooth vector bundle

This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some ...
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2answers
75 views

How to show $\exp(tX)\exp(tY)=\exp(t(X+Y)+tR(t))$ with $\displaystyle \lim_{t\to 0} R(t)=0$?

Let $X\in GL(n, \mathbb R)$. The exponential of $X$ is the matrix given by $$\exp(X)=\sum_{n=0}^\infty \frac{X^n}{n!}.$$ I need some help for showing the following result: ...
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26 views

Derivative of exponential map

Somehow I've gotten myself confused trying to take the derivative of the exponential map on $\mathfrak{so(3)}$. For vector $\theta$, $\delta \theta$, and $p \in \mathbb{R}^3$, define $$R(\theta, p) ...
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Theorem about the subgroup of a Lie group fixed by an involution

When trying to do Lie-theoretic calculations on Lie groups (finding the Bruhat decomposition, etc.) I've often come across expositions that seem to be implicitly using a result something like the ...
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2answers
62 views

Isometry group, one parameters sub-group geodesics

I have general questions about the group of isometries of a metric space. -When is the isometry group of a space a lie group? -when the isometry group is a Lie group, is there a relation between the ...
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To check d^2= 0 in the standard complex of Lie superalgebras.

For a Lie superalgebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $V$ we can define the cohomology $H^i(\mathfrak{g}, V)$ with coeffiecient in $V$ to be the cohomology space of the following complex: ...
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1answer
52 views

Matrix Lie algebras

I gave an answer to Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$? which was not popular. Meanwhile, i found myself at a loss when wishing to explain why a matrix Lie group had, ...
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Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$?

Is there a non-trivial subgroup $H \subset SL(2,\mathbb{R})$ such that $H \supset SO(2,\mathbb{R})$ ? My intuition is that, since $\dim SO(2)=1$ and $\dim SL(2)=3$, there should be some group ...
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23 views

Spin Representations and Galois correspondence?

I have a vague question regarding the Spin representations. Is there a "quick" way of seeing that $Spin(2n)$ has exactly two irreducible representations which do not factor through $SO(2n)$, and one ...
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1answer
15 views

How to show that $\alpha_{i_p}(s_{i_{p-1}} \cdots s_{i_1}(h)) = (s_{i_1} \cdots s_{i_{p-1}}(\alpha_{i_p}))(h)$?

Let $i_1, \ldots, i_p$ be integers and $\alpha_i$ be simple roots and $s_i$ be simple reflections in a Weyl group of type A. I checked some examples and it seems that $$\alpha_{i_p}(s_{i_{p-1}} \cdots ...
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25 views

What are the non-linear representations of $SO(3,1)$?

The classification of the representations of the Lorentz group $SO(3,1)$ is well known, but the representations are usually expressed in linear form. My question is whether there is a framework to ...
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24 views

Dimension of maximal tori

Let $G$ be a compact Lie group. $T$ $-$ its maximal torus. Is there a simple reasoning to show that dimensions of $T$ and $G$ have the same parity? I am sorry if this quesion is for children, but ...
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26 views

How does Fulton and Harris establish that the differential of a group hom respects ad?

Fulton and Harris, Representation Theory, Section 8.1 (pages 104 - 107 in my copy) is concerned with showing that group homomorphisms $\rho : G \to H$, where $G$ is connected, are completely ...