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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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 ...
4
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27 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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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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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
43 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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1answer
41 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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12 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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2answers
93 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
29 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
29 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 ...
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1answer
37 views
+50

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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39 views

$GL(n,\mathbb R)$ 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 ...
3
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1answer
24 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= ...
3
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5answers
161 views

Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} ...
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0answers
39 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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0answers
7 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 ...
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2answers
29 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 ...
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1answer
43 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
28 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
7 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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0answers
7 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 ...
3
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2answers
66 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 ...
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0answers
20 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 & & ...
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1answer
34 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
82 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
47 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
22 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 ...
0
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0answers
23 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 ...
3
votes
2answers
56 views

Differential of the inversion of Lie group

Let $G$ be a Lie group and $\iota \colon G\to G$ denote the inversion. If $e$ is the identity of $G$, prove that: $$\text d \iota _e = -\text {id} _{T_e G}.$$ I understand that the differential at $e$ ...
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0answers
12 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 ...
2
votes
1answer
268 views

Calculating the lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$ where this is defined as: $SO(2,1=\{X\in Mat_3(\mathbb{R})|X^t\eta X=\eta, \det(X)=1\}$ where $\eta$ is the matrix defined as: $$\left ...
0
votes
1answer
17 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
32 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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votes
2answers
53 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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2answers
66 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
votes
0answers
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 ...
0
votes
1answer
245 views

Lie algebra of the bounded continuous functions

I can think of the set of bounded, continuous functions from $\mathbb R \to \mathbb R$ as a group, with composition as addition of functions. In other words, this group has the rule that the ...
2
votes
1answer
254 views

What physical meaning do the dimension of Wigner d-matrices have?

Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also ...
0
votes
1answer
336 views

Map from $SL(2,\mathbb C)/SU(2)$ to $SL(2,\mathbb C)$

I am trying to write a map $s$, such that $s: SL(2,\mathbb{C})/SU(2) \to SL(2,\mathbb C)$. Using Iwasawa decomposition, I see that any element $g$ in $SL(2,\mathbb{C})$ can be written as $g = su$, ...
2
votes
1answer
183 views

Lie Group Multiplication in Coordinates

I'm having a bit of trouble with the last bit of Problem 3.2 in Kirillov Jr.'s Introduction to Lie Groups and Lie Algebras. (3.2) Let $f: \mathfrak{g} \rightarrow G$ be any smooth map such that ...
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1answer
32 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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1answer
25 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 ...
4
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1answer
42 views

Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
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Orthochronous Lorentz is time preserving and $\operatorname{SL}(2,R)$

Let's consider the pseudosphere in $\mathbb{R}^{1,2}$ given by $x^2+y^2-z^2=-R^2.$ We know that the group $O(1,2)=\{ A \in \operatorname{Mat}(3,\mathbb{R}): A^tGA=G \}$ where ...
2
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0answers
43 views

Transitive topological action

Let $G$ be a topological group, $A$ be set and $\mu\colon G\times A\to A$ a transitive action. I'm trying to prove the statement below is false. There exists only one topology in $A$ such ...
9
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1answer
73 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 ...
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2answers
47 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 ...
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1answer
43 views

Riemannian symmetric pair $(G,H)$ with H non-compact

Let $G$ denote a connected Lie group and $H$ a closed subgroup. Suppose that $\sigma$ is an involutive automorphism of $G$. Assume that $(G,H,\sigma)$ is a Riemannian symmetric pair. So far I have ...
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1answer
24 views

Lie algebra for SO(3) as a skew symmetric matrix

How can I show that the associated lie algebra for SO(3) is the set of all 3 dimensional skew-symmetric matrices?
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44 views

Embedded Lie subgroups are closed.

This is Exercise 2.1 from Kirillov's Lie theory book. Let $G$ be a Lie group and $H$ a closed Lie subgroup. Show that the closure $\overline{H}$ of $H$ in $G$ is closed in $G$. Show ...