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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The space of minimal geodesics on $SU(2m)$

In the proof of Bott periodicity for the unitary group in Milnor's Morse theory (Lemma 23.1, page 128), it is asserted that the space of minimal geodesics from $I$ to $-I$ in the special unitary group ...
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19 views

The irreducibility of representations of parabolic induction over $\mathfrak{gl}_n(\mathbb{C})$

I have the following problem: Let $\mathfrak{g} = \mathfrak{gl}(n)$ be the general linear Lie algebra over $\mathbb{C}$. Denote $\Pi = \{\alpha_i:=\epsilon_i-\epsilon_{i+1}\}_{i=1}^{n-1}$ by the ...
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9 views

Does the projection from a compact Lie group to its component group split?

This is an elementary question that probably admits an elementary counterexample, but ... Let $G$ be a compact Lie group and $G_0$ its identity component. One then has a short exact sequence $$ 1 \to ...
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30 views

Upper triangular matrices $UT_n(\mathbb K)$ is a matrix group: is my proof correct?

I am solving some exercises in Tapp's matrix groups for undergraduates and would be very grateful if someone could check my work (it's exercise 4.12): A matrix $A\in M_n(\mathbb K)$ is called ...
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22 views

Some solvable Lie algebra but not nilpotent

Can someone provide two concrete examples the Lie algebra which is solvable, but not nilpotent? -- And further explain the subtle differences between the solvable Lie algebra and the ...
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31 views

Intuitive understanding of lie group definition

So I have the following definition from the book: Definition: A matrix Lie group is any subgroup $G$ of $GL(n, \mathbb{C})$ with the following property: If $A_m$ is any sequence of matrices in $G$, ...
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57 views

Does every group have a lie algebra? [on hold]

Does the complex number group {$C, *$} have a lie algebra? When does a group have a lie algebra?
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21 views

Normalizer of normalizer of maximal torus in a Lie group

I'm stuck at this problem, Let $G$ be a compact connected Lie group and $T$ a maximal torus. Let $H$ be a closed subgroup of $G$. Let $N(T)$ and $N(H)$ denote the normalizers of $T$ and $H$ ...
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60 views

Prove the manifold of SU(2)/U(1) is the 2-sphere.

I want to demonstrate that the manifold of $SU(2)/U(1)$ is a 2-sphere. In a text-book I've found this way of solution, where there are some unclear points. Let to be $g= a\mathbb{1} + i b_j\sigma_j$ ...
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57 views
+50

Why a tensor product of 2x2 unitaries cannot implement a 3x3 unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
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17 views

cohomology ring of punctured spaces [closed]

What is the cohomology ring (algebra) of $$ H^*(S^{1}\vee S^1;\mathbb{Z})? $$ $$ H^*(\vee_k S^n;\mathbb{Z})? $$ The punctured torus $$ H^*(\prod_n S^1-*;\mathbb{Z})? $$ and $$ H^*(\prod_n ...
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29 views

The bijection between central characters and linkage classes over a semisimple Lie algebra

I have a question about the modules over a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For a given $\lambda \in ...
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1answer
30 views

Non-integral blocks of category $\mathcal{O}$ over $\mathfrak{sl}_2$ are semisimple.

Hi: I have a problem as follows. Consider the category $\mathcal{O}$ of $\mathfrak{g}: = \mathfrak{sl}_2(\mathbb{C})$. Let $r\in \mathbb{C}$ but $r\notin\mathbb{Z}$. Let $s_\alpha$ be the simple ...
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1answer
26 views

Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)\times SU(2)$, or their Lie algebras

I have seen sources claim that $SO^+(1,3) \cong SU(2) \times SU(2)$, but have seen others claim that only their Lie algebras are isomorphic. Is it true that $SO^+(1,3) \cong SU(2) \times SU(2)$? If ...
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1answer
55 views

Can someone direct me to prove the following three claims about a rotation group $SO(3)$ and its Lie Algebra?

In class my prof made three claims about a group and its Lie algebra. I cannot find direct reference to these claims because they are delivered in verbatim (im not even sure if I have them jogged down ...
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22 views

Proving subgroups of the unitary group U(n) are Lie groups

The problem: Let $K$ be a complex $n \times n$ matrix such that $K$ and $K^\dagger$ is non-singular and $K + K^\dagger$ is positive definite. Show that the set of complex $n \times n$ matrices $G$ ...
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156 views

Are there algebraic subgroups of Lie groups which happen to be Lie groups on their own but not Lie subgroups?

Let $G$ and $H$ be Lie groups and $i:H\to G$ an injective group homomorphism, not necessarily continuous. Once $i$ is continuous, it follows easily that $i$ is differentiable and even an immersion. ...
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52 views

Is every complex Lie algebra a complexification?

I'm wondering if every finite-dimensional complex Lie algebra can be written as a complexification of a real Lie algebra. At the vector space level, clearly every $\mathbb{C}^n$ is a complexification ...
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24 views

conjugation of Lie groups and homotopy group

Let $G$ be a Lie group. Let $\phi, \psi \in \pi_n(G)$. Consider $\theta \in \pi_n(G)$ defined by $\theta(x):= \phi(x)\psi(x)\phi(x)^{-1} \in G$, where we use multiplication and inversion in $G$ in ...
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1answer
42 views

Why $U$ generates $G$ as Lie group?

In line 2 of the proof, why is their intersection non-empty?
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1answer
37 views

Left-invariant differential form on homogeneous space

I'm trying to learn by working through an example but I got stuck with the line of reasoning provided in the text. Let $G$ be the group of matrices $$g= \begin{bmatrix} 1 &0 &\log{x} \\ y ...
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30 views

Spin group Spin(4,1)

i'm interested in the spin group $Spin(4,1)$ wich correspond to the symplectic group $Sp(1,1)$. The only source that I could find about it was wikipedia (http://en.wikipedia.org/wiki/Spin_group). It ...
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Why do entries of Maurer-Cartan 1-form span space of left-invariant 1-forms

Suppose $G$ is a $k$-dimensional Lie group of $n\times n$ matrices of the form $[g_{ij}(x_1,...,x_k)]$ where $g_{ij}$ are smooth functions. As a follow-up to the question I posted recently, I now ...
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1answer
46 views

The injectivity of torus in the category of abelian Lie groups

HI: I have the following question: Definition: A Lie group $T$ is called a torus if $T\cong \prod_{1\leq i\leq k} \mathbb{R}/\mathbb{Z}$ for some $k\in \mathbb{N}$. ${\bf Question}$: Is it true ...
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1answer
36 views

$E_6$ lie algebra and its representation

I've just started learning about Lie theory (only just finished up to basic classification of semisimple lie algebras) and I've got the following questions: How do I show that the complex lie algebra ...
2
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1answer
99 views

Weaker definitions of Lie subgroups

A Lie group $H$ is called a Lie subgroup of a Lie Group $G$ if there is a map $i:H\to G$ which is (a) an injective immersion and (b) a group homomorphism. My questios are: What happens if we replace ...
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2answers
50 views

Lattices in $PSL_2(\mathbb{R})$

Can a lattice in $PSL_2(\mathbb{R})$ have a normal abelian subgroup? It looks to me that it doesn't, but where can I read a proof?
2
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1answer
51 views

How does Maurer-Cartan form work

I have seen similar post asking for interpretation of the Maurer-Cartan form, but I am still struggling to understand it, so let me try to work a specific example and pose a specific question. Let ...
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1answer
51 views

The Underlying Manifolds of the Special Unitary Lie Groups SU(n)

I want to find the underlying manifolds of Lie Groups $\mbox{SU}(n)$ for general $n$. $$ \quad $$ My lecturer told us last year that the only n-spheres that admit a Lie group structure are ...
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32 views

One parameter subgroups on Lie groups and Riemannian metric

I read that geodesics of a bi-invariant metric on a compact Lie group are the one parameter subgroups. In a general Lie group, is it possible to create a Riemannian metric by transporting the ...
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2answers
102 views

Every 1- or 2-dimensional compact, connected Lie group is abelian

I know that in the 1-dimensional case the conclusion follows immediately from the fact that there exists a maximal torus (which then has to be the group itself). In the 2-dimensional case, we also ...
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23 views

Orbit space of action of a subgroup of a Lie group on a separable metric space

I am stuck on this question. Let $G$ be a Lie group acting freely on a separable metric space $X$. Assume that the orbit space $X/G$ is Hausdorff. Let $H$ be a normal Lie subgroup of $G$. Is the orbit ...
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14 views

Fixed subgroups of Cartan involution

Does every maximal compact subgroup of a semisimple Lie group with finite center occur as the fixed subgroup of a Cartan involution?
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1answer
25 views

Affine transformation invariants and lie groups

Is it possible to generate geometric properties which are invariant under affine transformations? I'm trying to learn about lie groups and lie algebras with the example of the lie group of affine ...
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2answers
27 views

Covering groups of compact connected Lie groups is compact.

Hi: I have a question as follows: Is it true that the covering group of a compact, connected Lie group is also compact? Thanks very much!
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12 views

Commuting Lie groups

I need help with a diagram (16) in the following paper, http://arxiv.org/pdf/1108.2472v1.pdf The groups of diffeomorphisms are not commuting, but in ...
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13 views

Complexifying Lie group actions

In Atiyah-Pressley's paper Convexity and Loop Groups, it is claimed that the Bruhat cells $C_\lambda$ are invariant under the natural $S^1 \times T$-action and that the authors will prove this claim. ...
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22 views

Principal orbit type

I have trouble understanding the proof of Proposition 1.2.5 on p.17 in Audin's Torus Actions on Symplectic Manifolds: Let $G\curvearrowright M$ be a smooth action of a compact Lie group $G$ on a ...
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1answer
67 views

Help needed to understand statements about torus

I am having trouble understanding two statements: Let $A$ be an algebraic curve in $\mathbb{P}^2$ over $\mathbb{C}.$ Consider its normalization $$\pi: \hat{A} \to A.$$ If genus $g(\hat{A})=1,$ ...
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24 views

lie group and lie algebra: confusion about the tangent space

I am reading a document about performing 2D and 3D transformations in space specifically rigid body transformations which can be represented using Lie groups. For example, the rigid body ...
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1answer
25 views

Confused about classification of simple lie algebras

In the classification of simple lie algebra, I learnt that $\mathfrak{sl}(n + 1)$ has a root system $A_n$. For example http://stacky.net/files/written/LieGroups/LieGroups.pdf, page 82. But I found ...
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1answer
47 views

Compact Lie group with non discrete center?

Could someone please give me an example of a compact Lie group with non discrete center which is not just a product of a group with a torus?
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1answer
42 views

Tensor product over Lie group isomorphic to that over its Lie algebra

Hi: I have a question about the tensor product of representation of Lie groups as follows: Let $G$ be a connected, complex Lie group with Lie algebra $\mathfrak{g}$. Let $V$ and $W$ are ...
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1answer
28 views

Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
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100 views

Harmonic Analysis on the Affine Group

In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking ...
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62 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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1answer
102 views

Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ...
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1answer
36 views

Periodic geodesics in special unitary groups

Looking at the solution of the equation of motion for the action using a bi-invariant metric on a Lie group. If I take the lie group as $SU(2)$ the geodesics are always periodic because $SU(2)$ is the ...
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1answer
45 views

Is my proof that $\operatorname{Isom_n}{\mathbb R^n}$ is compact right?

I previously did this exercise: Prove that $\operatorname{Isom_n}{\mathbb R^n}$ is a matrix group. where $$\operatorname{Isom_n}{\mathbb R^n} = \left \{ \left ( \begin{array}{cc} A & ...
2
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1answer
43 views

The orthogonal group $\mathcal O_n (\mathbb K)$ is isomorphic to a subgroup of $\mathcal O_{n+1} (\mathbb K)$

I tried to do the following exercise: Prove that $\mathcal O_n (\mathbb K)$ is isomorphic to a subgroup of $\mathcal O_{n+1} (\mathbb K)$ The definition of $\mathcal O_n (\mathbb K)$ is ...