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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Relation between simple roots and fundamental weights.

Let $\alpha_1, \ldots, \alpha_n$ be simple roots of a semisimple complex Lie algebra. Let $\omega_1, \ldots, \omega_n$ be the fundamental weights. We have $$ \alpha_i = \sum_{s} k_s \omega_s, $$ for ...
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33 views

Proving a relation for representations of gauge groups

Let ${\cal G}$ be a Lie group - possibly disconnected. Let ${\mathfrak g}$ denote the corresponding Lie algebra. Let $R_k$ be a general unitary representation of ${\cal G}$ and $R$ be the adjoint ...
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17 views

$B$-action on $U$.

Let $G$ be an algebraic, $B$ Borel subgroup, and $U$ unipoent subgroup of $G$. For example, we take $G=GL_n$, $B$ the subgroup of lower triangular matrices, and $U$ unipoent upper triangular matrices. ...
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1answer
30 views

What's wrong with this trivial proof that every element of a compact Lie group is contained in a maximal torus?

The Lie groups book I'm reading (Knapp, Lie Groups Beyond an Introduction, page 255) goes to some trouble to prove that every element of a compact Lie group is contained in a maximal torus. Why isn't ...
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21 views

Prove that ${\frac{d}{{dt}}_{t = 0}}(\det ({e^{tX}})) = Tr(X)?$ [on hold]

How to prove $${\frac{d}{{dt}}_{t = 0}}(\det ({e^{tX}})) = Tr(X)?$$
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60 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
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39 views

Are there common inequivalent definitions of Cartan subalgebra of a real Lie algebra?

I'm confused about seemingly different notions of a Cartan subalgebra of a real semisimple Lie algebra, and I'm wondering if there are common inequivalent definitions. In the book Lie Groups: Beyond ...
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1answer
28 views

Need some help with this exercise: why $\operatorname{Aff_n}$ is not closed

I ran into trouble while working on this exercise: Prove that $\operatorname{Aff_n}{(\mathbb K)} \subseteq GL_{n+1}(\mathbb K)$ is a matrix group where $$ \operatorname{Aff_n}{(\mathbb K)} = \left ...
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1answer
32 views

These subsets of $O(n)$ are clopen

Please could someone check my work on this exercise (from a book I am reading). Thanks! Exercise: Prove that $SO(n)$ and $ O(n)^- = \{ A \in O(n) \mid \det(A) = -1 \}$ are both clopen in $O(n)$. My ...
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42 views

Proof that $GL_n, SL_n$ are not bounded

Please could someone check my work on this exercise (from book I am reading). Thanks! Exercise: Prove that $GL_n (\mathbb K)$ is non-compact when $n \ge 1$. Prove that $SL_n (\mathbb K)$ is ...
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43 views

representation $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$

Let $V_{m,\,n}$ denote the vector space of the homogeneous complex polynomials of degree $m$ in $n$ variables (under addition). Define a representation $\pi_{m,\,n}: \text{SU}(n) \to ...
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1answer
29 views

How to check if a representation of su(2) is irreducible

I have found a representation $\rho$ of the group $G=Su(2)$. I want to show that this representation is irreducible but I don't know how. Finding all invariant subspaces seems very difficult. I ...
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1answer
26 views

Confusion about change of basis matrix

This video here seems to suggest that if a vector $v = (c_1, \dots, c_n)$ is given with coordinates in some basis $b_1, \dots, b_n$ and $B$ is the matrix with columns $b_1, \dots, b_n$ then $Bv$ is ...
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1answer
44 views

Open (but not closed) subgroups of $GL_n$

The book I am currently reading states: "...as we will see later, non-closed subgroups [of $GL_n(\mathbb K)$] are not necessarily manifolds." Prompted me to think about open subgroups of $GL_n$: ...
3
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29 views

Sets of orthogonal matrices are bounded

I have already shown that $O(n), SO(n), U(n), SU(n)$ and $Sp(n)$ are closed. Now I want to show that they are bounded. But when I tried, I noticed I need a metric or a norm on these sets. But there ...
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60 views

permutation group, lie group

Let $S$ be any set, and denote by $G$ the collection of all subsets of $S$. For $A, B \in G$ let be $AB = (A - B) \cup (B - A)$. I know how to show that this set $G$, with this product operation is a ...
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33 views

Heisenberg group as a suspension

I'm working on Heisenberg group and I want to understand the suspension viewpoint. Let me be more precise. Let us denote by $\mathbb{H}^3(A)$ the set of matrix \begin{equation} \begin{pmatrix} 1 ...
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29 views

Decomposiom of the representation of $SU(N)$

Let $T$ be the "fundamental" representation (I mean the one in which the matrices representing the group elements are simply themselves) of $SU(N)$ group. I have \begin{pmatrix} SU(N-1)& 0\\ ...
3
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25 views

What is an example of a map not satisfying this rank condition?

Definition: Consider a Lie Group $G$ and a set of right invariant vector fields on $G$, denoted $\Gamma$. A point $y \in G$ is called normally accessible from a point $x \in G$ by $\Gamma$ if there ...
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39 views

Soft question: A good book for introduction to Lie group book

I am taking next semester introduction lie groups. I was wondering what do you guys think what book should I use for this course.
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1answer
17 views

Dense and integral zero.

Let $G$ be a compact Lie group and $u\in C^{0}\left(G\right) $. If $\int_{G} u\left( g \right)v \left(g \right)dg= 0$ for every $v\in V $, a subset which is dense in $C^{0}\left(G\right)$, then ...
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1answer
37 views

Are there “interesting” examples of complete finite-volume non-compact Riemannian manifolds that are not non-positively curved?

Moreover, it would be good that, if $M$ is such an example, its universal cover $\tilde M$ is "highly" symmetric, i.e. the group G of isometries of $\tilde M$ is "big" (for example, transitive, or a ...
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1answer
70 views

Could someone check my work on this exercise

I solved the following exercise, could someone please check my work? Exercise: Let $$ A = \left ( \begin{array}{cccc} 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 ...
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2answers
41 views

Is $O(2)$ really not isomorphic to $SO(2)\times \{-1,1\}$?

An exercise in a book I'm reading is to show that $O(2)$ is not isomorphic to $SO(2)\times \{-1,1\}$. The problem is, I don't believe the statement. Let me elaborate why: $O(2)$ consists of ...
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2answers
30 views

Isn't this $f$ always a group isomorphism

Consider the following exercise from a book I'm reading: If $n$ is odd show that $$ f: O(n) \to SO(n) \times \{1,-1\}, A \mapsto (A \operatorname{det}{A}, \operatorname{det}{A})$$ is an ...
2
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2answers
20 views

Product of two arbitrary elements in $O(2)$

An exercise in a book I'm reading asks to describe the product of two arbitrary elements in $O(2)$. I would like to solve the exercise but I got stuck. I know that an element in $SO(2)$ can be written ...
3
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1answer
61 views

Induced Lie group action on a tangent bundle $TG\times TM\to TM$ and an example concerning Adjoint action

Suppose I have a Lie group action $$ G\times M\to M, (g,m)\mapsto g\cdot m. $$ which is transitive on $M$, then the tangent functor $T$ induces a corresponding map: $$ TG\times TM\to TM, (\delta ...
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60 views

Show that G is a Lie group and find a adjoint representation for G

$$G = \{ A \in GL(2,R): AA^t = p^2I, p>0, \det A >0\}$$ Show that G is a Lie group and find the explicity expression for their elements. And find a adjoint representation for G. Hi, I tried to ...
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14 views

What is the Jacobian for Sim(3) lie group action on 3D points ? (4d homogenous points)

I am coding up Sim(3) constraint types for a factor graph, and need to calculate the jacobian of the Sim(3) group action on 3D points. I am following the guide on http://ethaneade.com/lie.pdf ...
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12 views

Topology of orthogonal groups when $n > 4$?

On Wikipedia I read that the topologies of $O(1)$ and $SO(1)$ to $SO(4)$ are known topological spaces. What about $O(n), SO(n), U(n)$ when $n>4$?
3
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1answer
30 views

Proof help: $SU(2)$ is a double cover of $SO(3)$

I am reading a proof that $SU(2)$ is a double cover of $SO(3)$. My source is this set of notes: http://www.damtp.cam.ac.uk/user/examples/D18S.pdf. The proof begins near the bottom of page 4. I have ...
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2answers
50 views

How can I show that these matrices don't commute

I want to show that $A\in O(2) \setminus SO(2)$ and $B \in SO(2)$ don't commute. To prove it I wrote $$ B = \left ( \begin{array}{cc} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta ...
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1answer
22 views

How can I show that this matrix is a flip

I am trying to show that $$ F = \left(\begin{array}{cc} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array} \right ) $$ is a flip about a line through the origin. What I ...
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15 views

Decomposition of SU(n) anticommutator

In $SU(N)$, the special unitary group, the algebra generators $T_a$ are hermitian and traceless. The structure constants are fixed with $[T_a,T_b]=i f_{abc}T_c$. In the fundamental representation of ...
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21 views

Closure relations of the cells in the Bruhat decomposition of the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
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16 views

Haar measure of an angle-distance ball in SO3

If for rotations $R_0$, $R_1$ we define the distance $d(R_0, R_1)$ to be the angle of $R_0 R_1^{-1}$ and given $r\in [0,\pi)$, what is the "volume" (normalised Haar measure) in $SO_3$ of the ball ...
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23 views

maps between suspension of complex projective spaces and special unitary groups

How to do the following question? I get totally lost... this question is given by the professor in our final exam paper.
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1answer
24 views

Why does the maximal compact subgroup of a Lie group inject into the compact form?

I've seen multiple sources state the following (without proof or reference), but I don't see why it's true. Let $G$ be a Lie group, and $G_u$ be a compact connected Lie group such that the ...
4
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1answer
26 views

Lifting Riemannian metrics on principal bundles

Given a principal bundle $\pi:M\rightarrow M/G$, there are natural maps $$\pi_{\mathcal{F}}:\mathcal{F}(M)^G\rightarrow\mathcal{F}(M/G)$$ ...
15
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1answer
125 views

Decomposing $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles, formula for all $n$?

$``$Let $G = \text{SU}(2)$, and let $V_n$ be the space of homogeneous degree $n$ polynomials in $\mathbb{C}[x, y]$. Decompose $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles.$"$ For ...
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2answers
58 views

$SU(2)$ acting by conjugation, decomposition into irreducibles

I am attempting past exam questions of the Cambridge Math Tripos. I know how to do the first few parts, which involves giving the irreducible representations of $U(1)$ and $SU(2)$. But I am not sure ...
2
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1answer
35 views

Identifying the orbit space of the unitary group $U(n)$ in the compact symplectic group $Sp(n)$

Let $Sp(n)$ be the compact symplectic group. Let $U(n)$ the unitary group, and $O(n)$ the orthogonal group. What is $Sp(n)/U(n)$? What is $U(n)/O(n)$? I obtain that ...
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1answer
49 views

Decomposition into irreducibles of representations of semisimple Lie groups.

Let $G$ be a connected semisimple Lie group and $\mathfrak{g}$ it's Lie algebra. Then $\mathfrak{g}$ is semisimple. Let $V$ be a finite dimensional representation of $G$. Viewing $V$ as a ...
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1answer
35 views

CW Structure of $SU$

I'm reading Switzer's Algebraic Topology and he mentions that $SU = SU(\infty)$ can be given a CW complex structure. He also says that this implies, by a theorem of Milnor's, that $\Omega SU$ has the ...
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23 views

all differentials collapse of the Serre spectral sequence

Let fibration $$ SO(n)\to SO(n+1)\to S^n, $$ consider the Serre spectral sequence of cohomology $(E^{*,*}_k,d_k)$, $k\geq 2$, $E^{p,q}_2=H^p(S^n;\mathbb{Z}_2)\otimes H^q(SO(n);\mathbb{Z}_2)$. How ...
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1answer
28 views

Tangent space of matrix group is a Lie subalgebra

In my lecture today, we were covering matrix groups and Lie algebras. My professor made the statement that given any matrix group $G$, the tangent space of the group at the identity $T_{e}G$ is a Lie ...
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2answers
32 views

Finding the tangent space of a subgroup

My professor set the following question and I have an answer, though would like someone with more experience to cast a critical eye over the details as I don't necessarily trust my result! Define the ...
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1answer
37 views

Elements in finite symmetry groups share a common fixed point

I am reading Tapp's intro to matrix groups for undergraduates. On page 46 he states the following theorem: For $X\subseteq \mathbb R^2$ if $Symm(X)$ is finite then it is isomorphic to $D_m$ or ...
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35 views

Question about Modular function in Haar measure

I'm reading the book "Basic Lie Theory" (http://guests.mpim-bonn.mpg.de/abbaspou/Lie-Book_verrouille.pdf) and I'm trying to understand the proof of Lemma 2.3.4 which states that: Let $G$ be a locally ...
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68 views

How is the periodic structure of SO(n) reflected to its lie algebra so(n)?

An element of $SO(n)$ represents an rotation so that it must have identity with $2\pi$-like additional rotation. On the other hand, the elements of lie algebra $so(n)$ construct an noncompact vector ...