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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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 ...
4
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2answers
79 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: ...
3
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29 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 ...
2
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2answers
73 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 ...
3
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1answer
70 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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170 views

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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27 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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28 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 ...
2
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1answer
41 views

Maps between Manifolds and Maximal Rank

I'm trying to prove a theorem from Olver's Applications of Lie Groups to Differential Equations. It's supposed to be an "easy" consequence of the Implicit Function Theorem but I honestly can't see ...
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2answers
142 views

Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups

I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in this Math Overflow thread, it is mentioned ...
2
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42 views

Is the exponential map of GL(n,C) holomorphic?

Let $GL(n, \mathbb{C})$ be the complex general linear (Lie) group consisting of all invertible complex $n\times n$ matrices, and $gl(n,\mathbb{C})\cong C^{n^2}$ be its Lie algebra. The exponential map ...
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1answer
55 views

Do all Lie groups admit deformation retracts onto compact subgroups?

The noncompact Lie group $SL(n, \mathbb{R})$ admits a deformation retract onto the compact Lie group $SO(n, \mathbb{R})$ via polar decomposition. Do all noncompact Lie groups admit deformation ...
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2answers
53 views

left regular representation of SU(2)

in Sepanski's book Compact Lie groups, he describes the representation theory of SU(2) as being isomorphic to $\mathbb{N}$ (SU(2) acts irreducibly on the (n+1)-dimensional space of homogeneous ...
3
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1answer
54 views

Inducing a surface area measure on $S^2$ from the Haar measure on $SO(3)$

I'm reading the book "Random Matrices: High Dimensional Phenomena" by G. Blower. There is an example that I've been struggled for a long time. For those who have access to the book, it's the Example ...
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0answers
24 views

Rotations of form $R_z(a)R_y(b)R_x(c)R_z(-a)R_y(-b)R_x(-c)$

Any proper rotation (in three dimensions) can be expressed using the Tait-Bryan (sometimes called improper Euler) angles in the form $$ R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_x(\psi) $$ where ...
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1answer
54 views

Is every unitary irreducible representation an induced reperesentation?

I have recently read about induced representations and I have the following perhaps naive question about them. Let $G$ be a finite or infinite (Lie) group. Can we construct all irreducible unitary ...
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1answer
33 views

How to find the Lie algebra of a specific subgroup of a product Lie group

My question is about finding the Lie algebra of a specific Lie group. Start with a Lie group $G$, with normal Lie subgroup $C \unlhd G$. Then define the following subgroup $\hat{G} \leq G \times G$: ...
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1answer
44 views

Can topological groups be smoothed into lie groups?

I've been thinking about this for the past couple of days and I'm really not sure of the answer.. By "smoothed" I mean that for any arbitrary precision we can find a Lie group which approximates the ...
5
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1answer
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$GL(n,\mathbb R)$ not isomorphic to $SL(n,\mathbb R)\times \mathbb R^\ast$ when $n>1$

If $GL(n,\mathbb R) \cong SL(n,\mathbb R) \times \mathbb R^\ast$, then the centers are isomorphic. When $n$ is even, this means that \begin{align*} \mathbb R^\ast &\cong \{\lambda I_n \mid ...
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107 views

How to show a matrix can't be written as exponential?

How can I show the matrix $$A = \left( \begin{array}{c c} -2 & 0 \\ 0 & -1 \\ \end{array} \right)$$ can't be written as $A = exp(a)$? I've tried to write A like $$A = \left( ...
3
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1answer
41 views

computing the Haar measure for O(n) and U(n) groups

My question is about how to compute the Haar measure for O(n) and U(n) groups. For example, for the conventional parametrization of SO(3) with 3 angels, the Haar measure is $ dO= ...
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0answers
46 views

How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
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2answers
18 views

How to show that $GL_n/U$ is birationally isomorphic to $B^-$?

It is said that $GL_n/U$ is birationally isomorphic to $B^-$. Here $U$ acts by right multiplication on $GL_n$. I think that $GL_n/U$ consisting of cosets. Two matrices in the same coset if any two ...
0
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1answer
39 views

Sympletic matrix must have a determinant equal to one. [duplicate]

I hope that I am just confused, but I don't see why a sympletic map must have a determinant equal to one and not minus one?-Could anybody help me with that?- I am referring to the group $$Sp = \{T ...
1
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1answer
52 views

Exercise 5.8 from Lie Group, Daniel Bump

In the exercise 5.8 Bump has asked to prove that the group $Sp(4)$ over complex numbers, which is usual complex embedding $U(4)\cap Sp(4,\mathbb{C})$, can be described by, $$\left\{\begin{pmatrix} ...
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1answer
31 views

How can we prove that $GL(2,\mathbb{R})$ is a topological group in $R^{4}$

A group is called topological group if it satisfies three properties 1) G is a Hausdorff space in K (Here we want to prove that if $ A,B \in GL(2,\mathbb{R})$ then we can find two disjoint open sets ...
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0answers
14 views

P-adic Lie groups - Representation theory

I am quite familiar with the Representation Theory for locally compact groups and nilpotent Lie groups. I want to start with the study of $p$-adic Lie groups representation theory, in particular ...
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15 views

Cayley-Hamilton type decomposition of SL(3,R) matrices

Given an element $\lambda = \theta_a T_a$ of SL(3,R) Lie algebra, where $T_a$s are the generators and $\theta_a$s are parameters, is there a general formula to determine the coefficients A,B and C ...
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24 views

prove or disprove that a particular invertible matrix is also orthogonal

is it true that, if for some $2n \times 2n$ matrices $O^t=O^{-1}$ and $$J_0= \begin{bmatrix} \begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & ...
4
votes
1answer
51 views

If a connected Lie group is divisible, is its exponential map surjective?

A group $G$ is divisible if for all $g \in G$ and $k \in \mathbb{Z}_+$ there is an element $h \in G$ such that $h^k = g$; we call such an $h$ a $k$th root of $g$. In an answer to a recent question ...
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2answers
88 views

Is there a theory of lie “rings”?

A Lie group is a group that is a differentiable manifold and addition and inversion are differentiable maps. Is there a theory for rings that are differential manifolds and have differentiable ...
1
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1answer
56 views

set of almost complex structures on $\mathbb R^4$ as two disjoint spheres

The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ ...
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0answers
24 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
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25 views

Isometries of S^3 and some Lie algebras

By considering $S^3$ as the group of unit quaternions, and letting it act on itself from both the left and right, one can get an isomorphism $SO(4)\cong (S^3\times S^3)/C_2$, where the $C_2$ subgroup ...
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17 views

Element of the spin group

I've got the following question: why it true, that if I have a unitary element $u$ in the (real) Clifford algebra $Cl(V,g)$ which is even and the operator $\varphi(u)$ defined via ...
3
votes
2answers
72 views

T//W for adjoint type group PGL3

Let $G$ be a reductive algebraic group and $T$ a maximal torus (over $\mathbb{C}$). It is well known that if $G$ is simply connected type then $T//W = \mathbb{A}^r$. I want to verify that the ...
1
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1answer
48 views

Factorization of Compact Lie Algebras into Irreducible Ideals

I have read in some lecture notes on Lie theory that any compact Lie algebra $\mathfrak{g}$ can be factored as a direct sum of of irreducible ideals for the $\mathrm{ad}$ representation. That is, ...
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1answer
24 views

Algebra of a cartesian product of two or more copies of a Lie Group

Let $G$ be a Lie Group end let us consider the Cartesian product of $n$ copies of $G$ ($N\geq 2$): $G\times G\times \ldots \times G$. What is the Lie Algebra of this group? Is the Lie Algebra the ...
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2answers
61 views

Correspondence between one-parameter subgroup and left-invariant vector field.

Given a one-parameter subgroup of a Lie group $G$, we can show that the one-parameter subgroup $\phi : \mathbb{R} \to G$ defines a unique left-invariant vector field $X: \frac{d\phi^\mu(t)}{dt} = ...
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2answers
69 views

How to compute dimension of $O(n,\mathbb{R})$

Let $f:GL(n,\mathbb{R})\to GL(n,\mathbb{R})$ be the smooth map $A\mapsto A^TA$. Observe that $f$ has constant rank on $GL(n,\mathbb{R})$ by chain rule and that $O(n,\mathbb{R})$ is the preimage of ...
4
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1answer
73 views

Visualizing Lie algebra of SO(3)

Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix: $$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ ...
2
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15 views

Determining Matrix Group from Connected Component

I am interested in finding a method for determining all the matrix subgroups of a matrix group that have a specific connected component. This is what I thought would work from what I have read so far ...
3
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1answer
37 views

How to obtain a Lie algebra homomorphism from a Lie group homomorphism

In class we learn a theorem tells us one can cook up a Lie algebra from a Lie group: If $f: G\to H$ is a homomorphism of a Lie group then $T_I f: T_I G\to T_I H$ is a homomorphism of Lie algebra. ...
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81 views

Understanding Dynkin Diagrams - any organising ideas - are they now adequately understood?

Some 30 or so years ago JH Conway posed a question about the ubiquity of the Dynkin Diagram - not necessarily in public, but I heard him ask it. I think it was in the context of "what would be ...
2
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2answers
48 views

Exercise on left invariant 2-forms

I'm trying to solve a problem about the Lie group of transformation over $\mathbb{R}$: \begin{equation} x\mapsto ax+b, \end{equation} where $a,b\in\mathbb{R}, a\neq 0.$ I'm asked to find the space of ...
2
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1answer
27 views

Maximal tori in $SO(n,\mathbb{C})$

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$) Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in ...