# Tagged Questions

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 ...

0answers
57 views

### Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
1answer
29 views

### maximal tori and principal $N(T)$-bundles.

Let $U(n)$ be the unitary group and $T^{n}= S^{1} \times \cdots \times S^{1}$ a maximal torus in $U(n)$. Let $N(T^{n})$ be the normalizer in $U(n)$ of $T^{n}$. How can i prove that \$U(n) \rightarrow ...
1answer
70 views

### $U(n)/U(n-1)$ as homogeneous space

How can I prove that the quotient $U(n)/U(n-1) \simeq S^{2n+1}$ (where $U(n)$ is the unitary group). Is il correlated with the teory of homogeneous spaces?
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98 views

### why the orthogonal group $O(k,l)$ is homotopy equivalent to $SO(K)\times SO(l)$

I want to prove that the orthogonal group $O(k,l)$ (http://en.wikipedia.org/wiki/Indefinite_orthogonal_group)is homotopy equivalent to $SO(k)\times SO(l)$, so that ...
0answers
40 views

### Covering space, Weyl group, flag manifold.

Let $G$ be a compact Lie group and $T$ a maximal torus in $G$. We define the Weyl group $W$ as the quotient space ${N_{G}}(T)/T$, where ${N_{G}}(T)$ is the normalizer of $T$ in $G$. We ...
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66 views

### Covering space (Lie groups and their maximal tori)

Let $G$ be a compact Lie group and $T$ a maximal torus in $G$. We define the Weyl group $W$ as the quotient space ${N_{G}}(T)/T$, where ${N_{G}}(T)$ is the normalizer of $T$ in $G$. We ...
1answer
32 views

### a neighbourhood of identity U generates G where g is a connected lie group

Let G be a connected Lie group and U any neighbourhood of the identity element. How to prove that U generates G.
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28 views

### Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
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103 views

### An example of a Lie group

I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a 2D blanket or a circle/curve or a ...
1answer
58 views

### Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?
1answer
64 views

### Defining Lie groups without the notion of a manifold

I like to introduce (matrix) Lie groups without the notion of manifolds. (And I am happy to scarify the "few" Lie groups which are not matrix groups in favor of a simpler definition.) I was thinking ...
1answer
60 views

### Dimension of Lie algebra according to root system

I was wondering how is it possible to find the dimension of a semi-simple lie algebra $L$ if its corresponding root system is (lets make it simple) of type $B_2$. We can find the number of roots and ...
2answers
60 views

### Nilpotent Lie Group that is not simply connect nor product of Lie Groups?

I have been trying to find for days an non-abelian nilpotent Lie Group that is not simply connect nor product of Lie Groups, but haven't been able to succeed. Is there an example of this, or hints to ...
1answer
34 views

### Differential action on a complex manifold

Let $M$ be a complex manifold of dimension $n$. Furthermore assume that we have a action of a Lie-Group $G$ on $M$ i.e. $G \times M \rightarrow M$, which is differential, meaning that for every \$g \in ...
1answer
76 views

### Invariant inner products on infite-dimensional representations

Let $G$ be a compact group and let $V$ be it's continuous representation. It is well known that if $V$ is finite-dimensional, then there is an $G$-invariant inner product on $V$. I haven't found a ...
2answers
128 views

### When does the $\mathfrak g$-invariance of the symplectic form imply $G$-invariance?

Let $G$ be a Lie group with Lie algebra $\mathfrak g$, and let $M$ be a smooth manifold. Suppose $G$ acts on $M$, $G \to \text{Diff}(M)$. This naturally induces an action $\mathfrak g \times M \to M$ ...
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29 views

### Is it possible to further simplify the product of three exponentials $e^A e^B e^C$ when $[A,C]=kB$ (k is a scalar)

The background is calculation of the little group elements of Poincare group for massless particles. I start with a bunch of exponentials of operators, and the end goal is to crunch them into the ...
1answer
47 views

### Nilpotence of Lie Algebra

I am trying to show that if $L$ is a Lie algebra and $L/Z(L)$ is nilpotent than $L$ is also nilpotent. Can someone please help me? I tried to first show by induction: $(L/Z(L))^k=L^k/Z(L)$. Is it ...
1answer
184 views

### Geometric interpretation of the map $SO(4) \to SO(3)$

Let me first explain the background of my question. As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
1answer
198 views

### Finding All Irreducible Representations of $SO(3)$

I've read that one may prove that all irreducible representations of $SO(3)$ are tensor product representations of the fundamental representation (or tensor product representations of the spin 1/2 ...
2answers
207 views

### $SU(2)$ Representation of $SO(3)$

I've often seen it written that $SU(2)$ is a "two-valued representation" of $SO(3)$ (in theoretical physics books mainly). I have a major conceptual issue with this however. I know there is a Lie ...
2answers
145 views

### Lie group and SO3 visualisation

Maybe I'm asking a very vague question but I'd like to know if there are some visualisation tools available already that explain lie algebra exponential map or logarithm? I'd like to be able to ...
1answer
47 views

### Schur's first lemma for finitely generated continuous groups of $SU(d)$

Suppose that a finite set $S$ of $d\times d$ special unitary matrices densely generates a representation $\rho$ of a continuous subgroup of $G$ of $SU(d)$. That is, for every $\epsilon>0$ and ...
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34 views

### How to find the induced Lie algebra homomorphism

Consider the quaternions $H=\{1+bi+cj+dk, a,b,c,d \in \mathbb{R}\}$ and the norm $\|h\|=\sqrt{h^*h}$, which is a Lie group homomorphism between $H^*$ and $\mathbb R^*$. How can I find the Lie algebra ...
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42 views

### Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
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44 views

### How to prove that a lie group is simply connected

I need to prove that $O(3,19)/SO(2)\times O(1,19)$ is simply connected. In particular $O(n_{+},n_{-})$ denotes the orthogonal group of $\mathbb{R}^{n_{+}+n_{-}}$ endowed with the diagonal quadratic ...
1answer
62 views

### Quaternions as group of rotation and scaling

It is very well known that unit quaternions are well suited to represent rotations in 3D. In particular, the group of unit quaternions forms a double cover of the special orthogonal group $SO(3)$. ...
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27 views

### Dynkin diagrams

Let $\gamma$ be a tripple edged graph that is associated with an admissible set in a real inner product space. Please, how do I show that $\gamma$ is the Coxeter graph of the Dynkin diagram, $G_2$?
1answer
38 views

### Lie algebra 3 Dimensional with 2 Dimensional derived lie algebra #2

I read the book of Karin Erdmann and Mark Wildon: "An introduction to Lie algebras". In page 22 they say that: If $\dim (L) = 3$, $\dim (L') = 2$ then (a) $L'$ abelian and (b) \$\operatorname{ad} x ...
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60 views

### Does the $O(n)$ bundle of a manifold depend on the metric?

Let $g_1$ and $g_2$ be two Riemannian metrics on a manifold $M$. These induce two $O(n)$ bundles on $M$, whose fibers over each point $x\in M$ are the groups of orthogonal transformations of $T_x M$ ...
0answers
31 views

### Detail in polar action

I am reading a paper "Tits geometry and positive curvature - Fang, Grove, and Thorbergsson" See the following site http://arxiv.org/pdf/1205.6222.pdf In page 7, the 9-th line from the bottom ...
1answer
77 views

### establishing an isomorphism

I request help with this is a question from Introduction to Lie algebra by Erdmann and Wildon. The question asks to show that show that \$so(4,\mathbf{C})\cong sl(2,\mathbf{C}) \oplus ...
1answer
104 views

### $3\times 3$ symmetric matrix with signature $(2,1)$

I need to show the set of $3\times 3$ real symmetric matrices with signature $(2,1)$ is an open connected subset in the usual topology of $\mathbb{R}^6$. To show connectedness I did like the ...
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63 views

### Product of all rotation matrices in $\mathrm{SO}(3)$

With curiousity, I'm trying to prove whether multiplication of all rotation matrixes in $\mathrm{SO}(3)$ is identity irrelevant of multiplication order. As each rotation matrix in $\mathrm{SO}(3)$ ...
1answer
33 views

### How can I show that $ASL_n(F)$ is acting 2-transitively?

One of my friends asked me to ask this question here. This is a question from his last exam: Let ASL_n(F)=\{T_{A,v}:V_n(F)\to V_n(F)\mid\exists A\in SL_n(F), \exists v\in V_n(F), ...
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36 views

### Smooth Action of a Finite Group

Suppose $H$ is a finite group acting smoothly on a smooth connected manifold $M$. The action is trivially proper, as $H$ is discrete. If the action of $H$ were also known to be free, i.e. \$h\cdot ...
1answer
34 views

### Examples of Pansu differentiable maps

I would like to know some non-too-trivial examples of Pansu-differentiable maps between stratified groups (real ones, not $\mathbb{R}^n$, pun intended). For example, can anyone name a ...
2answers
68 views

### Elements of finite order in compact abelian Lie Group

If $G$ is a compact abelian Lie group, why does the $n$th power map from $G$ to $G$ form a finite covering? I cannot see why the kernel must be finite.
1answer
71 views

### Isometries from Diffeomorphisms

Given a compact, connected Lie group G of diffeomorphisms on a manifold M, how to construct a Riemannian metric on M such that elements of G are isometries of M?
1answer
261 views

### Given a group $G$, how many topological/Lie group structures does $G$ have?

Given any abstract group $G$, how much is known about which types of topological/Lie group structures it might have? Any abstract group $G$ will have the structure of a discrete topological group ...
2answers
75 views

### SO(5)-invariant metrics on the 4-sphere

Are there any examples of Riemannian metrics on $S^{4} \subset \mathbb{R}^{5}$ that are not SO(5)-invariant? Or are all metrics on the 4-sphere SO(5)-invariant? Hope my question is not too trivial ...
1answer
50 views

### Symmetry, change of variables

I am having trouble understanding a section in these notes. It is on page 3. Section 3 -- Discretization of the Korteweg-de Vries equation. I don't understand why $$V_4=x∂_x+3t∂_t-2u∂_u$$ generates a ...
1answer
67 views

### Quotient group $S^3/\{+I,-I\}$

How can I prove that the quotient group $S^3/\{+I,-I\}$ is isomorphic to $SO_3$ and that the group $S^3$ is not isomophic to $SO_3$? Here $S^3$ is the subgroup of the quaternion group: ...
1answer
55 views

### Error in Weyl character formula computation.

I need someone with a keen eye for errors. I am trying to use the Weyl character formula for the symplectic group Sp$(4,\mathbb{C})$ on certain matrices coming from 2x2 quaternion matrices. Summing ...
1answer
66 views

### Dimension of the GL-orbit of d-forms in one less variable

Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
0answers
68 views

### Submanifold of a Lie group - tangent space

Let $G$ be a compact Lie group and $H, H' \leq G$ Lie subgroups. Consider the set $M = H' \cdot H = \{h\cdot h' \ \vert \ h \in H, h' \in H'\}$. Is it possible to describe explicitly the tangent space ...
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35 views

### Group of affine transformation in plane is unimodular

I am trying to do an exercise in the book "Analysis on Lie group" as follows: Let $G$ be the group of all affine transformations in the plane, i.e. $G$ contains all the mapping of form \$(x,y)\mapsto ...
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124 views

### Integrating angular velocity to obtain orientation

Suppose that $\gamma:[0,1]\to \operatorname{SO}(3)$ is a path in the space of orientation preserving rotations of $\mathbb R^3$. It is classical that we can find a corresponding \$\omega:[0,1]\to ...
1answer
57 views

### A question about Lie algebras corresponding to Lie groups and algebraic groups

Lie groups and algebraic groups both correspond with Lie algebras, which are by definition the left invariant vector field. But the topology of Lie groups and algebraic groups are different. Are their ...
2answers
129 views

### Lie algebra 3 Dimensional with 2 Dimensional derived lie algebra

i read in mark wildon book , an introduction to lie algebras, in page 22 say that : Suppose that dim $L$ = 3 and dim $L'$ = 2. We shall see that, over $\mathbb{C}$ at least, there are infinitely many ...
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