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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1answer
180 views

What's so special about unipotent groups

Why are they so important? I see them appear in Lie theory, algebraic geometry, etc. Can somebody elaborate? For example, can someone explain why they are such natural groups to consider in ...
3
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0answers
99 views

lattices in semisimple Lie groups

I would like to learn more on lattices in semisimple Lie groups, especially their relations with Coxeter groups. Does anyone have suggestions of books that could be useful? Thanks!
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0answers
190 views

$SU(2)$ is a covering space of $SO(3)$.

The method of topology is very clear.Then there's a question asking to use adjoint representation of lie group $SU(2)$ $(\operatorname{adj}:SU(2)\to GL(su(2)))$to prove this. I can't solve this .
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2answers
147 views

Is the multiplicative complex plane a Lie group?

I know that the complex plane is a Lie group with +, but is it also a Lie group with the usual complex multiplication? This would give us a nice geometrical interpretation of the famous Euler ...
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0answers
95 views

Can the infinite von Dyck groups be subgroups of $SU(n)$?

I know by constructing some particular cases that I can find unitary matrices $X$, $Y$ and $Z$ such that $X^m = Y^n = Z^p = XYZ = 1$ with $$ \frac{1}{m} + \frac{1}{n}+\frac{1}{p} < 1 $$ ...
3
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93 views

About the realization of $SO(N)$ given by Daniel Bump in his book Lie Group

Thank for your interest for my question and thank you very much if you can answer me. In his book p.187, Daniel Bump says that a realization (a representation) of $SO(2n)$ is given by the unitary ...
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1answer
73 views

Why does rigidity hold only if rank >1?

In simple words, why does Margulis' superrigidity and arithemiticit only hold for lattices in Lie groups of rank $>1$? E.g. what is the reason for it to fail for $SL(2,R)$?
3
votes
1answer
274 views

Conditions for left-invariant one-forms to be closed.

Let $G$ be a connected (semisimple) Lie group with Lie algebra $\frak{g}$. For $\omega \in \frak{g}^*$, we may define a left invariant one-form on $G$ by $\left[ \omega (g)\right] (v)=\omega \left( ...
3
votes
1answer
137 views

How do I find all the roots of a lie algebra and hence its root system diagram given the cartan matrix for that algebra?

If I am given the cartan matrix, I can find the $2\langle αi , αj\rangle/\langle αj , αj\rangle$ of the simple roots where $α$ are the simple roots. But, from this how do I find the $\langle αi , ...
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0answers
51 views

$G_1$-Scalar factors for Clebsch-Gordan coefficients for $ U(n)$

when evaluating the $G_1$ scalar factors for CGC's of $U(n)$ it seems that some of the factors are undefined. The explicit formula for the evaluation of the scalar factors is Eq. (6) in 18.2.8 of N.J. ...
2
votes
2answers
43 views

Showing $R(G) = R(T)^W$

Let $G$ be a compact connected Lie group and $T$ a maximal torus. Let $R(G)$ be the representation ring of $G$. Then restriction of reps gives a map $R(G) \to R(T)^W$, where $R(T)^W$ are the ...
18
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1answer
387 views

Can we prove that there are countably many isomorphism classes of compact Lie groups without appealing to the classification of simple Lie algebras?

It is a nontrivial fact that there are only countably many isomorphism classes of compact Lie groups. One can prove this by a series of reductions: first to the connected case, then to the simply ...
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0answers
37 views

Subspace of a homogeneous space.

Given two homegenous spaces $\frac{G}{H}$, $\frac{A}{B}$ with $A\subset B$ is there a way to prove that $\frac{A}{B}\subset \frac{G}{H}$ ie that $B\subset A\cap H$? In particular I would like to ...
5
votes
1answer
408 views

Subgroups of a vector space

I would like to have an overview of how a subgroup of a vector space over $\mathbb R$ of dimension $n$ can look like. Is there a complete classification available? I know that there are for examples ...
4
votes
0answers
140 views

Non-closed subgroups of Lie groups

The following are my impressions from playing with a line of irrational slope $\gamma$ in the standard torus $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$, $\mathbb{R}\hookrightarrow ...
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vote
2answers
712 views

Cayley Transform, Exponential Mapping and more…

Assume a self-adjoint operator, represented as hermitian matrix $H=H^\dagger$. To my knowledge there are at least 2 mappings of $H$ onto unitary matrices: Cayley's Transformation with ...
4
votes
2answers
343 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 ...
3
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0answers
57 views

An unexplained iso between $H^{m+1}(O(m+2)/O(m), S^m)$ and $H^{m+1}(S^{m+1})$

I am reading topology of Lie groups by Mimura and Toda and got to the part where they are beginning to compute $H^*(O(n))$, page 120. If we let $r_m :S^m \to O(m+1)$ be the map that sends $v$ to the ...
2
votes
1answer
105 views

The systematic method for the explicit construction of representations of su(2) algebra? (e.g. pauli matrices)

How do I construct the su(2) representations of a given dimension?
4
votes
0answers
170 views

internal direct product of lie groups

If $G$ is a (edit: simply connected)Lie group, when does a direct sum decomposition of its Lie algebra (into a direct sum of subalgebras) correspond to a (semi)direct product decomposition of $G$? ...
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votes
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 ...
4
votes
2answers
190 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 ...
1
vote
1answer
457 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
893 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 } ...
6
votes
1answer
811 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 $G$. ...
1
vote
1answer
216 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 ...
1
vote
1answer
170 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 ...
4
votes
1answer
496 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 ...
1
vote
0answers
65 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 ...
1
vote
1answer
87 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?
3
votes
3answers
354 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 ...
2
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0answers
81 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 ...
1
vote
1answer
125 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 ...
0
votes
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
1
vote
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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votes
2answers
383 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 ...
5
votes
1answer
211 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 ...
3
votes
1answer
454 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 ...
2
votes
0answers
219 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 ...
6
votes
1answer
72 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 ...
3
votes
2answers
219 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 ...
2
votes
0answers
225 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 ...
2
votes
0answers
90 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
votes
2answers
475 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.
4
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2answers
472 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 ...
0
votes
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
votes
1answer
108 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 ...
0
votes
1answer
149 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 ...
2
votes
1answer
139 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 ...
2
votes
1answer
167 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 ...