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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2answers
119 views

Is this a one dimensional Lorentz Boost? And can you have a 1-d Boost without group structure?

Someone has claimed that he has constructed a quaternion representation of the one dimensional (along the x axis) Lorentz Boost. His quaternion Lorentz Boost is $v'=hvh^*+ 1/2( ...
2
votes
1answer
119 views

Cartan decomposition of measure of $GL(2, \mathbb{C})$

I am searching for a integration formula $$ \int\limits_{GL_2(\mathbb{C})} \phi(g) d g = \int\limits_{SU(2)} \int\limits_{SU(2)} \int\limits_{M} \phi(k_1mk_2) \omega(m) d m d k_1 d k_2 ,$$ where $M$ ...
0
votes
2answers
312 views

Is this proof that SU(2) cannot be isomorphic to SO(1,3) valid?

It seems intuitively obvious to me that there cannot be an isomorphism between $\mathrm{SU}(2)$ and $\mathrm{SU}(2)\times\mathrm{SU}(2)$ where SU(2) is the Lie Group with the Pauli matrices as ...
4
votes
1answer
400 views

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
2
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0answers
82 views

How do I construct the $\operatorname{SU}(2)$ representation of the Lorentz Group using that $\text{SU}(2)\times\text{SU}(2)\cong \text{SO}(3,1)$?

This question is so mathematical that I think I'll have more luck asking it in the mathematics section, than I would in the physics section. This is problem II.3.1 in Anthony Zee's book Quantum Field ...
6
votes
0answers
262 views

Root Systems of Lie Groups.

Let $G$ be a compact Lie group assumed to be a subgroup of $U(n)$. Also, let $T$ be a maximal torus of G. Then there exists a basis $\{v_1, \ldots ,v_d\}$ of the Lie algebra of $G$, $\mathfrak{g}$, ...
2
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0answers
92 views

Finding Lie Subgroups

I've been asked to find all proper lie subgroups of $SU(2)$. I seem to recall thinking that $U(1)$ is the only nontrivial connected lie subgroup, but I can't quite remember how I came up with this ...
2
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0answers
103 views

Reference Request - Spaces of Smooth Vectors

I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
4
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0answers
305 views

Bi-invariant form on compact connected Lie group

Let $G$ be a Lie group and $\omega$ be a left invariant $k$-form, how to prove that $r^*_a \omega$ is left invariant? What I do: $(l^*_g (r^*_a \omega))_x (v_1, \ldots,v_k)=(r^*_a \omega))_{gx} ...
5
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2answers
255 views

$SU(2)$ Lie group

I have been studying Lie groups for a bit of fun for a while now and think they are fascinating. I have recently been told that $SU(2)$ can be used in some way to keep track of navigational systems in ...
3
votes
2answers
397 views

$SO(3)$ Lie group

I'm a little stuck at the moment where to go next with this. I know that there is a fact that there is a curve in $SO(3)$, beginning and ending at the identity which cannot be deformed to the constant ...
5
votes
1answer
96 views

Agreement of two polynomials

How can I prove that if two polynomials (of matrix coefficients) agree in a neighborhood of $0$, then they are identical? Thank you!
1
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0answers
107 views

Continuous but not uniformly continuous in $\text{GL}(2,\Bbb{C})$

Essentially, I'm trying to find an example of a function that is continuous but not uniformly continuous on $\text{GL}(2,\Bbb{C})$. I'm aware that this group is isomorphic (up to constant multiples) ...
14
votes
6answers
526 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
6
votes
1answer
247 views

A compact Lie group has descending chain condition on closed subgroups.

Proposition: Let $G$ be a compact Lie group and let $$G\supset G_1\supset G_2\supset\ldots$$ be a chain of closed subgroups $G_i$ of $G$. Then this chain must eventually stabilize. Question: The hint ...
2
votes
1answer
67 views

$\operatorname{Isom}{(M)}$ has Lie-structure for M metrizable manifold

Suppose $M$ is a smooth and metrizable manifold. Then $\operatorname{Isom}{(M)}$ can be given the structure of a Lie group, so that the action of $\operatorname{Isom}{(M)}$ on $M$ is still smooth. I ...
2
votes
1answer
599 views

What is the Lie algebra of the ``indefinite orthogonal group''?

For $p,q \geq 0$ and $n=p+q\geq 1$, give $\mathbb{R}^n$ the indefinite inner product (written as a matrix) $$ \begin{pmatrix} I_p & \\ & -I_q \end{pmatrix}, $$ where $I_m$ is the $m \times m$ ...
2
votes
0answers
67 views

Basis of the Engel algebra

If I have a connected, simply connected nilpotent lie group given by the commutators between the elements of a basis of its Lie algebra how can I recover the left invariant vector fields? For ...
3
votes
0answers
97 views

why this happens ? (dilation)

let be the dilation operator $ x \frac{d}{dx} $ i know this operator is invariant under the change $ y=ax$ for any positive 'a' real number however let be the change $ y= \frac{-1}{x}$ then the ...
2
votes
0answers
102 views

Different definitions of the complex Grassmannian

I have come across two different definitions of what I suspect is the same object. Both are called the complex Grassmannian: 1: $U(n)/U(k)\times U(n-k)$ 2: $SU(n)/(S(U(k)\times U(n-k))$ What is the ...
1
vote
0answers
50 views

Are tempered representations unitarizabile?

Let $G$ be a locally compact, unimodular group and $Z$ be its center Clearly, square integrable representations with central unitary character is unitarizabile, since their matrix coeffecient imbed ...
6
votes
2answers
1k views

About connected Lie Groups

How can I prove that a connected Lie Group is generated by any neighborhood of the identity? The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this ...
3
votes
2answers
505 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
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vote
1answer
134 views

Conjugate exponential integral formula for Lie algebra

Somewhere in my notes the following formula appears $\int_0^1 e^{s R} \frac{d R}{dt} e^{(1-s)R} ds = \frac{d}{dt} e^{R}$ where $R$ depends on $t$, and has values in a Lie algebra [$\mathfrak{so}(3)$ ...
2
votes
1answer
96 views

$SL(3,\mathbb{C})$ acting on Complex Polynomials of $3$ variables of degree $2$

So I'm given the following definition: $h(g)p(z)=p(g^{-1}z)$ where g is an element of $SL(3,\mathbb{C})$, $p$ is in the vector space of homogenous complex polynomials of $3$ variables and $z$ is in ...
0
votes
2answers
349 views

Maximal Abelian Subgroup of SL(3,C)

So I'm looking to find the maximal abelian subgroup of SL(3,C). I know that if a maximal torus for SL(3,C) exists that said torus is the maximal abelian subgroup. Is it enough to know that since ...
21
votes
2answers
347 views

How can I lift a path to $\mathrm{Spin}(n)$?

Suppose I am given an explicit differentiable path $\gamma\colon[a,b]\to SO(n)$, with $\gamma(a)=\gamma(b)=I$. Then $\gamma$ either does or does not lift to a closed loop in $\mathrm{Spin}(n)$. How ...
4
votes
1answer
352 views

Why is $Sp(2m)$ as regular set of $f(A)=A^tJA-J$, and, hence a Lie group.

I'm trying to prove that $Sp(2m)$ is a Lie group using this: defining a function $f(A)=A^tJA-J$ and trying to see that this is a submersion. But I've not realized yet what is the domains and the range ...
1
vote
1answer
164 views

The Lie Algebra of a trivial Lie Group

I was wondering if it is true that the lie algebra of a trivial lie group is trivial? Surely the answer is yes but I just want to make sure.
1
vote
0answers
190 views

The connected Lie subgroups of a n-dimensional torus.

I am stuck on the following question Let $T^n$ be a $n$-dimensional torus. What are the connected Lie subgroups of $T^n$? Which of these are closed subgroups? Does anyone have any ideas on how to ...
2
votes
2answers
293 views

Fourier transform over Lie group

let be the Lie Group of translations $ y=x+a$ and dilations $ y=bx $ whose generator are $ \frac{d}{dx} $ and $ x\frac{d}{dx} $ then could i define the Fourier transform over this group if i use a ...
2
votes
2answers
282 views

Lie Group homomorphisms

I am revising for my Lie Groups exam and am stuck on the following question. Find all Lie Group homomorphisms a) $ \ F : \mathbb{R} \longrightarrow S^1 \ $ (Hint: Consider the corresponding ...
2
votes
0answers
65 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
1
vote
1answer
102 views

The simplest way to prove that any left-invariant vector field on a Lie group is complete

It's all in the question: I look for the most intuitive proof that the integral curves of any left-invaraint vector field on a Lie group can be extended for all values of "time". I realize that the ...
4
votes
1answer
132 views

Adjoint action smooth (in Tapp's book)?

I'm reading "Matrix Groups for Undergraduates" by Tapp with a student. A "matrix group" means a subgroup $G$ of $GL_n(\mathbb R)$ which is (relatively) closed-- so if $(A_n)\subseteq G$ with ...
8
votes
2answers
748 views

Universal Cover of $SL_{2}(\mathbb{R})$

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
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0answers
51 views

Performing integration over $U(d)$

Is there any more or less efficient way to integrate a function (not necessarily a polynomial) over $U(d)$?
1
vote
0answers
78 views

What is $D_{5}$

I have recently encountered a Lie group in a paper called $D_{5}$ as a subgroup of $E_{6}$. I have tried googling with mixed results. Is it just $\operatorname{SO}_{10}(\mathbb C)$?. Is it ...
2
votes
1answer
104 views

Classifying Lie group homomorphisms

I'd like to know whether there exists a standard way of classifying homomorphisms between two given Lie Groups, at least for some class of Lie groups, e.g. matrix groups. For instance, suppose that I ...
6
votes
2answers
287 views

Reference request: GL(n)

There are many places, which describe the unitary irreducible representations of $GL(n, F)$ with $F = \mathbb{C}$ or $F =\mathbb{R}$. Basically, we obtain a bunch of parabolically induced ...
6
votes
0answers
153 views

Finding the universal cover of a matrix group

Consider the group $G = \left\{\begin{pmatrix} a & b\\ 0 & 1\end{pmatrix}: a \in \mathbb{C}^{\times}, b \in \mathbb{C}\right\} \subset GL(2, \mathbb{C})$. How does one find the universal cover ...
1
vote
0answers
77 views

Dilation Invariance

Given the formula $$F(x)= \sum_{n=-\infty}^{\infty}f(x+n) $$ We know that is invariant under translations of the form $y=x+n$ for any integer $n$. However can we find a similar formula for dilations ...
4
votes
1answer
414 views

Can Young tableaux determine all the irreducible representations of Lie groups?

Can Young tableaux, or generalisations thereof, determine and parametrise (uniquely) all the irreducible representations of each simple Lie group over the complex numbers, ignoring the 5 exceptions? ...
1
vote
1answer
133 views

Basic Lie Algebra Question

Essentially, I'm trying to prove that when computing the tangent space for a group that there's nothing special about considering it at only the identity. Namely, there is an isomorphism of vector ...
6
votes
1answer
371 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
7
votes
1answer
2k views

The special orthogonal group is a manifold

How can we show that $SO(n)$ is an $n^2$-manifold. It would be tempting to say that $SO(n)$ is an open set of $\mathbb R^{n^2}$ but this is not the case since $SO(n)$ is given as the intersection of ...
3
votes
0answers
351 views

Continuous map of Lie Groups

This might be a dumb question, but are all continuous maps between Lie groups also homomorphisms? I can only seem to think of examples in which they are (i.e., $GL(n,\mathbb{R}) \to \mathbb{R}$ via ...
0
votes
0answers
86 views

Maslov Index product property.

I am not sure how to show the following property of the Maslov Index, which in McDuff and Salamon's Introduction to Symplectic Topology, Theorem 2.35, is called the product property. Let $\Lambda: ...
11
votes
5answers
344 views

$\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$

I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$. At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
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0answers
269 views

Writing down cohomology groups of the complex Grassmannian

I am studying a homogeneous space and would like to know its cohomology groups. Using some sequences and fibrations I have figured out some of these groups, but largely in terms of the cohomology ...