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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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
19 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
25 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

$GL(n,\mathbb R)$ 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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86 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( ...
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1answer
23 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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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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6 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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26 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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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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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 ...
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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
35 views

PSL(2,C) and PSL(2,R) [closed]

What is PSL(2,C) transformation and PSL(2, R) transformation? and what is the difference between them? And what is the transformation matrix for three given points in each PSL(2,C) and PSL(2,R)? For ...
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1answer
32 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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81 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 ...
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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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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 ...
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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 ...
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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 ...
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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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35 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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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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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
65 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 \\ ...
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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 ...
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1answer
31 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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72 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
46 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
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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2answers
55 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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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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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 ...
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24 views

What do the right and left translation maps on $GL_n(R)$ look like?

Yesterday I asked this question: What does $GL_n(R)$ look like? (And got some good answers.) Since I would like to have some models of noncommutative lie groups in my head, I was wondering if anyone ...
3
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2answers
117 views

What does $GL_n(R)$ look like?

Exactly as in the title - what does the general linear group "look like" (you are free to interpret this however you like) as submanifold of $R^{n^2}$? What should I imagine when I think of it? (I ...
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1answer
12 views

Commutative lie groups - how is $(R, >, 1)$ a $T^q \times R^p$

I just found out that the connected, commutative lie groups are all products of the form $T^q \times R^p$, where T is the circle and R the real numbers. Is the set of positive reals under ...
4
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0answers
55 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
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35 views

SAGE vs. Mathematica for Lie algebras / groups?

What math software is better for working with Lie algebras and Lie groups, SAGE or Mathematica?
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1answer
30 views

Implied relationships between Lie groups and Lie algebras.

Suppose $\mathcal{L}$ is a finite-dimensional Lie algebra, and $\mathcal{G} = e^{\mathcal{L}}$ is it's compact, connected Lie group. Given a closed sub-algebra $\mathcal{L}' \subset \mathcal{L}$, it ...
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32 views

How to calculate the Maurer-Cartan form in the adjoint representation?

While I am reading a paper, I come across a difficulty. Here, we have a Lie group and we know its Lie algebra defined as $[G_a,G_b]=f_{ab}^{\phantom{ab}c}G_c$ with $G_a\in\mathfrak g$. Under the ...
2
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2answers
44 views

Is the convolution operation some kind of group operation?

I'm just curious but will the convolution operation be any sort of group operation? A motivating example would be to see that the natural exponential family of distribution functions are closed under ...
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1answer
95 views

Almost all subgroups of a Lie group are free

I am currently reading this paper by Epstein. I need help with understanding the proof. Specifically, I have the following two questions. Let $w\colon G\to H$ be an analytic mapping between ...
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15 views

Convergence of Baker-Cambpbell-Hausdorf for compact groups

It is well known that the Baker-Campbell-Hausdorf formula doesn't need to converge for general elements of a Lie algebra, resp. for matrices with norms larger then 1. On the other side, if $G$ is a ...
3
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0answers
31 views

How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
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17 views

Affine connection defined by a quotient manifold?

Suppose $G$ is a Lie group with affine connection $X,Y \mapsto\nabla_X Y\in C^{\infty}(G,TG)$, and $Q$ is a subgroup of $G$ such that $G/Q$ is also a nontrivial Lie group. Does this quotient manifold ...
3
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12 views

Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ I can approximate this to first order as: $$\tilde ...
3
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1answer
50 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)