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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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 ...
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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 ...
4
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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?
3
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2answers
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 ...
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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 ...
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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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2answers
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 ...
2
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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 $S_{n}$?
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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 ...
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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 ...
3
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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 ...
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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 ...
3
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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 ...
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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 $\mathfrak g \times M \to M$ ...
4
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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 ...
3
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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 ...
15
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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 ...
2
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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 ...
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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 ...
2
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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 ...
2
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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: ...
2
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0answers
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 ...
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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, $G_2$?
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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 $T_x M$ ...
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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 ...
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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 ...
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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 $\mathrm{SO}(3)$ ...
2
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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 ...
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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 ...
2
votes
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.
5
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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?
11
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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 ...
3
votes
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 ...
4
votes
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 ...
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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: ...
1
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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 ...
5
votes
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 ...
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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 ...
3
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0answers
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 ...
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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 ...
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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 ...
