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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51 views

What is the weight of an operator?

What does this mean: "The standard Cartan weight operators for SO(4) are $L_{12}$ and $L_{34}$. An SO(4) irrep is labeled by the highest weight defined by these operators, which is of the form ...
34
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7answers
3k views

How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?

I've noticed that $\mathrm{GL}_n(\mathbb R)$ is not a connected space, because if it were $\det(\mathrm{GL}_n(\mathbb R))$ (where $\det$ is the function ascribing to each $n\times n$ matrix its ...
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0answers
112 views

How can I generate $\mathrm{SL}(n,\mathbb Z)$ by the subgroup $\mathrm{SL}(n-1,\mathbb Z)$ and another Element of $\mathrm{SL}(n,\mathbb Z)$?

Let $\{z_1,...,z_n\}$ be the canonical Basis of $\mathbb{Z}^n$, such that $z_i$ equals the vector $(0,\dotsc,0,i,0,\dotsc,0)$ with a 1 in the $i$th position. I want to show that the ...
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2answers
184 views

A trivial question concerning $sl_{n}\mathbb{C}$ representations

The question is, does the fact $$ \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 0 & 1 &0 \end{array}\right)^{2}=0, \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 ...
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1answer
412 views

Proof: Matrix exponential maps from tangent space to Matrix (Lie) group

Let us assume we have a definition of the tangent space (e.g. as in Proof: Tangent space of the general linear group is the set of all squared matrices). Furthermore, we already verified that the ...
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2answers
761 views

Proof: Tangent space of the general linear group is the set of all squared matrices

Let us assume we have the following definition of a tangent space: Definition of smooth path Let $X\subset\mathbb{R}^n$. Let $I$ be a real interval. \begin{equation} P \text{ is a smooth path in } ...
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1answer
718 views

Orbit-stabilizer theorem for Lie groups?

Let $G$ be a finite-dimensional lie group, with a transitive action on the points of a smooth finite-dimensional manifold $S$. Let $p$ be some point of $S$ and let $T$ be the stabilizer of $p$ in $G$. ...
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1answer
207 views

Lie Brackets of Nilpotent Lie Algebras

Suppose I have the Heisenberg group H say over the $p$-adic integers $\mathbb{Z}_p$, which is the set of $3\times 3$ uni-upper-trianglar matrices over $\mathbb{Z}_p$ . Its Lie algebra $h$ is the set ...
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1answer
168 views

$SO(3)$ acting on the space of $3 \times 3$ matrices

Let $SO(3)$ act on the space of $3 \times 3$ real matrices by conjugation. How can I decompose the space of matrices into the sum on minimal invariant subspaces and figure out what they are isomorphic ...
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1answer
447 views

Various actions of the Weyl group

Using the notation of my previous question, let $N(T)$ denote the normalizer of the maximal torus $T$ and hence the Weyl group $W(G,T) = N(T)/T$. Here think of roots $\alpha$ as maps $T \rightarrow ...
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63 views

Maximal compact subgroup of SO(16) and SU(11)

The (compact real forms) Lie groups $SO(16)$ et $SU(11)$ are different, but have the same dimension ($120$). The (compact real form) Lie group $SO(11)$, of dimension $55$, is an obvious sub-group of ...
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1answer
86 views

Centralizers in reductive Liegroups = unimodular?

Let $G$ be a real reductive group. Why is the centralizer of an element unimodular? What is a reference?
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3answers
304 views

Classification by Dynkin diagram: Why's there no $E_9$?

from Wiki According to Dynkin's classification, we have as possibilities these only, where n is the number of nodes: To me it seems not obvious why there should not be a $E_9$? Further clicking ...
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0answers
79 views

Minimal embedding of exceptionnal Lie groups into special orthogonal groups

Let $G$ be a Lie group. The set of all $N$ such that $G$ is a subgroup of $SO(N)$ has a minimum $N_{\min}(G)$. (If I am not wrong, $N_{\min}(G)$ is supposed to be less or equal to $\dim(G)$) What is ...
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1answer
123 views

Finding determinant of matrix lie group

Given $G=\left\{A\in M_2(\mathbb{R})\mid A^\top XA = X\right\}$ where $X = \pmatrix{3&1\\1&1}$ and a Lie algebra $\mathfrak g=\left\{Y\in M_2(\mathbb{R})\mid Y^\top X+XY = 0\right\}$, how ...
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1answer
130 views

General form of a Lie Algebra

Given $G=\left\{A\in M_2(\mathbb{R})\mid A^\top XA = X\right\}$. Need to find the basis. Error in question
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2answers
147 views

Getting a Lie algebra from a Lie group.

Given $G=\left\{A\in M_2(\mathbb{R})\mid A^\top XA = X\right\}$ where $X = \pmatrix{3&1\\1&1}$. Let a smooth path $A(t)$ in $G$ with $A(0)=I_2$.
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2answers
345 views

How to show that the unitary group $U(n)$ is not isomorphic to the semidirect product $U^n(1)\rtimes S_n$?

I came to this problem when doing the exercise that the polydisc $\Delta(0,1)^n=\prod\limits_{n}\Delta(0,1)$ in $\mathbb{C}^n$ is not biholomorphic to the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$.We ...
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1answer
196 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
401 views

Universal enveloping algebra as algebra of differential operators

Let $G$ be Lie group and $g$ be its Lie algebra. Is it true (and if not generally, then under which circumstances) that the the algebra of its differential operators is isomorphic to the universal ...
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0answers
215 views

What is the constant $e$, fundamentally? [duplicate]

Possible Duplicate: Why is the number e so important in mathematics? Intuitive Understanding of the constant “e” The number $e$ is important in many respects. If you ask ...
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1answer
71 views

Relationship between $\operatorname{Ad}$ and $\operatorname{ad}$

This is kind of a question about Lie groups/algebras, but what is really hiding is some combinatorial work and some linear algebra In the context of Matrix Lie groups we can define the ...
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2answers
203 views

Why is a Lie Group simply connected iff it is simple and connected?

Yesterday i heard that a connected Lie group $G$ is simply connected (i.e. $\pi_1(G)=0$) if and only if $G$ is simple as a group, i.e. $G$ has no nontrivial normal subgroup. That sounds too good to be ...
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0answers
219 views

How do I derive the Poincaré group and its Lie algebra?

The relativity group of Minkowski spacetime is the subgroup $P < Aff(4,\mathbb{R})$ which preserves the proper time $c^2 (x_4-y_4)^2 - \|\mathbf{x}-\mathbf{y}\|^2$ between two events ...
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0answers
82 views

symplectic representations: when could the center act trivially?

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
5
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2answers
445 views

The fundamental group of Lie group

If $G$ is a compact Lie group whose Lie algebra $g$ has a trivial center, please show that the fundamental group of $G$ is finite.
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2answers
422 views

Classification of unitary irreducible representation

I recently learnt that one can explicitly classify the unitary irreducible representations of $\mathrm{SL}(2,\mathbb R)$. In the end one has a list of all these representations given by explicit ...
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1answer
48 views

$O(3)$ after identifying certain rotations

suppose i have $O(3)$ as a group and then proceed to identify rotations on the same axis. That is, assuming an element in the simple component is written as $$ e^{s_i I_i } $$ where $I_i$ are ...
0
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1answer
107 views

Lie group multiplication/Parameter space

So I have a set P={ $p(\alpha,\beta,\gamma)=\pmatrix{1&\alpha&\beta\\0&1&\gamma\\0&0&1}$ $|$ $\alpha,\beta,\gamma$ $\in R$}. I needed to show P is a Lie group, which I have ...
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1answer
144 views

Maximal tori in U(2)

I'm attending a course about Lie Groups, and in an exercise I'm asked to "find the maximal abelian subgroup of $U(2)$". Certainly an abelian subgroup of $U(2)$ (and in fact of any $U(n)$ increasing ...
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30 views

$(g^{-1}dg)^3$ for SO(p,q) group

It is well known that for SU(2) $(g^{-1}dg)^3$ can be expressed as a total differential d(...). However in physics literature I could not find similar expressions for the orthogonal groups. Is ...
2
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1answer
133 views

The Grassmanian as a Homogenous Space and Related Spaces

I am interested in studying a quotient of Lie groups related to the Grassmanian. I don't know very much topology so this question will be a little bit open ended. Let $p \neq q$ and consider the ...
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0answers
147 views

Special functions as representations of Lie Groups

-The spherical harmonics $Y_{lm}$ are complete on $L^2(S^2)$. They are also a representation of the (compact) Lie group $SO_3 (\mathbf{R})$. -The functions $e^{i n x}$ are complete on ...
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1answer
483 views

Centers of quotients of Lie Groups

Exercise 7.11 in Fulton's Representation Theory asks to prove that: (a) Show that any discrete normal subgroup of a connected Lie group $G$ is in the center $Z(G)$ (b) If $Z(G)$ is discrete, show ...
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1answer
352 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 $g = su$, ...
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1answer
263 views

Complete reducibility of sl(3,F) as an sl(2,F)-module

I was reading the Weyl's theorem on the complete reducibility of a finite dimensional representation of semi-simple Lie algerba and wanted to apply the theorem to the following problem which was ...
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1answer
182 views

SO(n) bi-invariant metric

Prove that the induced metric on SO(n) is bi-invariant. The inner product is given by the Frobenius inner product on matrices.
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3answers
532 views

Proper, smooth action with singular orbit space

This is a problem from Lee, Smooth Manifolds (Problem 9.4). It's not an homework problem, I'm just systematically solving every problem of that book, and I got stuck on this one. Usually I try not to ...
2
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1answer
106 views

Form of isomorphisms between subgroups of GLn

Suppose that I've been given two subgroups of the general linear group (over the real numbers), which are isomorphic. From this information alone, can I deduce the form of the isomorphism? I suspect ...
2
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2answers
76 views

Do there exist families of groups $G_{s}$ such that $\forall s\in[0,1], |G_{s}|=\mathfrak{c}$

Do there exist families of isomorphism classes of groups $G_{s}$ such that $\forall s\in[0,1], |G_{s}|=\mathfrak{c}$, where $\mathfrak{c}$ is the cardinality of the continuum, and for any ...
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1answer
535 views

Special orthogonal group as a manifold

Knowing that $\mathrm{O}(n,\mathbb{R})$ is a closed submanifold (of the general linear group) and that $\mathrm{SO}(n,\mathbb{R})$ is one of its subgroups with the same dimension, is there a quick way ...
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0answers
81 views

Does triality survive in product Lie groups?

Look at the following diagrams of Lie groups ...
2
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3answers
239 views

Lie algebra of $GL_n(\mathbb{C})$

I would like to do Tao's exercise 6 (i) but before I can even attempt it I need to be clear about his terminology. Exercise 6 Show that the Lie algebra $gl_n(\mathbb{C})$ of the general linear group ...
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2answers
402 views

“Change of basis” from standard vector space to matrix Lie algebra, and its inverse

For matrix Lie groups, the exponential map is usually defined as a mapping from the set of $n \times n$ matrices onto itself. However, sometimes it is useful to have a minimal parametrization of our ...
9
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1answer
211 views

Lie Groups which are not Hausdorff

I suspect this isn't a terribly difficult question, but I don't know the answer and I'd guess someone has already looked into it. Is it possible for a Lie group on a non-Hausdorff manifold to exist? ...
10
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6answers
3k views

Jacobian matrix of the Rodrigues' formula (exponential map)

I am working an algorithm which is supposed to align a pair of images. The motion model, which describes the pose $p$ of an image (with respect to the second) in 3D space, is purely rotational. ...
3
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1answer
456 views

Derivative of function including matrix logarithm

Is the following equation a first order approximation or incorrect for general matrix Lie groups? And what are the higher order terms? $$\frac{\partial}{\partial\mathbf x} (\log(\mathtt ...
7
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1answer
604 views

De Rham Cohomology of a Lie group

If $G$ is a connected Lie group with Lie algebra $\mathcal{G}$, then de Rham cohomology of left invariant différential forms $H_L^*(G)$ is isomorphic to the Chevalley–Eilenberg cohomology ...
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1answer
197 views

Lie group structure on some topological spaces

I have some basic background in Lie theory and I have some difficulties to show that some topological spaces admits a Lie group structure. More precisely, for a given Lie group $G$: 1) Why its ...
3
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1answer
537 views

First and second homotopy groups of a connected Lie group

I try to understand why for a connected Lie group $G$ the first fundamental group $\pi_1(G)$ is abelian, and mainly why the second fundamental group is trivial $\pi_2(G)=0$? Thanks for anyone who ...