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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37 views

Matrix multiplication in quaternions is not necessarily linear

I tried to show by example that matrix multiplication for quaternionic matrices is is not necessarily $\mathbb H$-linear. If $A \in M_n(\mathbb H)$ is a quaternionic matrix and $x$ is a vector in ...
3
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1answer
102 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
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34 views

Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
3
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1answer
45 views

Subgroup of $GL_{n+m}(\mathbb K)$

Let $G$ be a subgroup of $GL_n(\mathbb K)$ and $H$ a subgroup of $GL_m(\mathbb K)$ where $\mathbb K \in \{\mathbb R, \mathbb C, \mathbb H\}$. I want to prove that there exists a subgroup of ...
2
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0answers
24 views

Association of financial phenomena/indications with the conservation laws of Black Scholes equation

For a while I've been doing research on methods of obtaining conservation laws via the symmetries of DEs. I'm presently doing research on identifying financial indicators/phenomena that can be ...
3
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3answers
77 views

Why not $SL_n (\mathbb R)$ in this exercise

I just solved the following exercise: Let $SL_2(\mathbb Z)$ denote the set of $2\times2$ matrices with integer entries and determinant $1$. Prove that $SL_2(\mathbb Z)$ is a subgroup of ...
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0answers
31 views

$SO(n)$ is connected, alternative form

I have the following exercise: Show that $SO(n)$ is connected, using the following outline: For the case $n = 1$, there is nothing to show, since a $1\times 1$ matrix with determinant one must be ...
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1answer
45 views

Adjoint Lie algebra homomorphism

I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
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0answers
23 views

Question on inner product on space of representations of compact Lie groups

Let $K$ be a compact connected Lie group, wiewed as subgroup of unipotent matrices. Let $G=\mathfrak{k}^\mathbb C$ be the complexification with Lie algebra $\mathfrak{g}=\mathfrak{k}\oplus ...
2
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1answer
52 views

Cohomology of classifying spaces of exceptional Lie groups

Given an exceptional Lie group $G$ and maximal torus $T$ thereof, the inclusion $T \hookrightarrow G$ induces a map $BT \to BG$ of classifying spaces and a cohomological pullback $$H^*(BG) \cong ...
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35 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
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1answer
182 views

A question on the unit tangent bundle of the sphere and $SO(3)$

Let the unit tangent bundle be defined as follows: $$T^1S^2=\{(p,v)\in \mathbb R^3 \times \mathbb R^3 | |p|=|v|=1 \text{ and } p \bot v \}$$ Let $SO(3)$ be the group of rotations of $\mathbb R^3$. ...
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1answer
60 views

Group Axioms Motivation

Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several ...
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1answer
65 views

Induced Lie algebra homomorphism from Lie group homomorphism: left translation

A general result of Lie Theory is that every Lie group homomorphism $\Phi: G\rightarrow H$ induces a Lie algebra homomorphism $\phi: \frak{g} \rightarrow \frak{h}$. Which Lie algebra homomorphism ...
8
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1answer
216 views

Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
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1answer
24 views

Generalization of Euler angle decomposition for unitary operators?

it is well known that a general operator from SU(2) can be effectively represented as a composition of at most three "one-parameter" ones. For instance in the form $\exp(i\, \alpha\, \sigma_z) ...
2
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2answers
91 views

Is the $G$-action on a principal $G$-bundle proper?

Let $G$ be a Lie group. If $G$ acts properly and freely on a manifold $P$, then it is well-known that $P \to P/G$ form a principal $G$-bundle. I would like to know the converse: namely Question: if ...
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1answer
36 views

Is a Lie group connected?

I want to know if a Lie group is connected in general situation.I also want some example of Lie group. I will appreciate your help.
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1answer
54 views

Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
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0answers
55 views

Exact sequences of $1 \to A \to SO(N) \to B \to 1$, special orthogonal group

Inspired by the nice post and this, apart from SU(N), now I am particularly looking into the exact sequences of SO(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A ...
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0answers
40 views

Exact sequences of $1 \to A \to SU(N) \to B \to 1$, special unitary group

Inspired by the nice post, I am particularly looking into the exact sequences of SU(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A \to SU(N) \to B \to 1$$ where ...
4
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0answers
55 views

Exact sequences of SU(N) and SO(N)

We know that the spin group $Spin(N)$ has a short exact sequence of Lie groups. $$1 \to Z_2 \to Spin(N) \to SO(N) \to 1$$ I wonder whether there are some examples for SU(N) and SO(N) group, such ...
3
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5answers
179 views

Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} ...
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0answers
40 views

A Question on integration formula on $KAK$ decomposition

The following proposition appears in page 141 in Knapp's book, representation theory of semisimple groups. Let $G$ be linear connected reductive, and fix a positive system $\Sigma^+$ of restricted ...
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0answers
35 views

with which binary action is the 2-sphere a Lie group?

show that the 2-sphere is a Lie group. I do not know with which binary action is the 2-sphere a Lie group? and with which binary action is the $\mathbb{R^3}-{0}$ a Lie group? 2-sphere is equal with ...
2
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1answer
149 views

What is the manifold structure of U(n)?

A Lie group is simultaneously a differentiable manifold. As I understand it, the Lie group is generated via exponentiation of the generators of the Lie algebra. It is intuitively clear to me that the ...
1
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1answer
64 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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0answers
36 views

What are the root systems for the n-dimensional torus?

My question may seem silly at first, but currently I am not able to work out the question of finding all roots for the n-dimensional torus. At first, it seemed obvious to me that there are no roots at ...
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0answers
32 views

Group operation in $SO(3)$ is well-defined

I've just started to read Tapp's Matrix groups for undergraduates and it says: "$SO(3)$ becomes a group under composition of motions (since different motions might place the globe in the same position ...
2
votes
0answers
63 views

Question on $\mathfrak{sl}(2,\mathbb R)$

I am confused about some facts on $SL(2,\mathbb R)$. The Lie algebra of $SL(2,\mathbb R)$ is $\mathfrak{sl}(2,\mathbb R)$. However, the map $$ \exp:\mathfrak{sl}(2,\mathbb R)\ \rightarrow ...
2
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0answers
42 views

Weyl Group of Parabolic subgroups

Let $G=SL(n,\mathbb R)$ with Lie algebra $\mathfrak{g}=\mathfrak{sl}(n,\mathbb R)$. The classical minimal parabolic subgroup $B$ consists of the upper triangular matrices. The parabolic subgroups $P$ ...
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0answers
17 views

Character of a symmetric square

Let $V$ be a representation of $\mathfrak{sl}_2(\mathbb{C})$. As far as I am concerned a character of $V$ is a Laurent polynomial $\sum_{k\in\mathbb{Z}}d_k\cdot t^k$, where $d_k$ is the dimension of ...
0
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1answer
21 views

Representation of $\mathfrak{sl}_2(\mathbb{C})$ corresponding to Lie algebra representation

We have a representation $R$ of a Lie group $\mathrm{SL}_2(\mathbb{C})$ in the space of polynomials $\mathbb{C}[x,y]$ such that $R\begin{pmatrix} a & b \\ c & ...
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0answers
22 views

Homogeneous Spaces: The Erlangen Programme

This is a wholly a question of mathematics history. The Klein Erlangen programme is most pithily, if a little tersely, described in modern wording as a homogeneous space: a topological group acting ...
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18 views

Questions about an action of $U(\mathfrak{g})$.

Let $\mathfrak{g}$ be a Lie algebra and $U(\mathfrak{g})$ its universal envoloping algebra. Let $G$ be the Lie group of $\mathfrak{g}$ and $U$, $B^{-}$ the upper unipotent subgroup and lower Borel ...
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0answers
22 views

What is $d\mu$ of $\mu:T^*\mathbb{C}^n\rightarrow \mathfrak{gl}_n^*$?

This is an elementary question. Let $\mu:T^*\mathbb{C}^n \longrightarrow \mathfrak{gl}_n^*$ be the moment map given by $(x,y)\mapsto xy$. Then concretely, what is the differential $d\mu$ of $\mu$? ...
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0answers
24 views

Real or complex representation

How can one know for a given algebra $\frak{g}$ if a specific representation is real or complex? For example if $\frak{g}=so(10)$ how can one know that the representation $\underline{16}$ is complex? ...
6
votes
2answers
219 views

Lie Groups/Lie algebras to algebraic groups

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on ...
3
votes
0answers
38 views

Compact Lie algebras and Lie groups

A simple or semisimple Lie algebra is said to be compact if the $\mathrm{Tr}\left \{ T^\mathrm{adj}_{a}, T^\mathrm{adj}_{b}\right \}$ is positive definite where $T^\mathrm{Adj}_{a}$ are the generators ...
3
votes
1answer
123 views

Which subgroup of $\mathrm{SL}(2,\mathbb{C})$ is this?

I am looking into sub-algebras of $\mathfrak{sl}_2(\mathbb{C})$ and the subgroups of $\mathrm{SL}(2,\mathbb{C})$ they generate. The basis of $\mathfrak{sl}_2(\mathbb{C})$ I am using consists of 3 ...
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0answers
22 views

Are there some examples of Kac-Moody groups which are not reductive?

Are there some examples of Kac-Moody groups which are not reductive? Thank you very much.
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0answers
13 views

How to prove that $U(\mathfrak{h})$ is isomorphic to $\mathcal{O}(\mathfrak{h}^*)$.

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. It is said that $U(\mathfrak{h})$ is isomorphic to $\mathcal{O}(\mathfrak{h}^*)$. Here $\mathcal{O}(\mathfrak{h}^*)$ is the ring of ...
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1answer
22 views

What is the Lie group of $\mathfrak{h}$?

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. What is the Lie group of $\mathfrak{h}$? By definition, the Lie group of $\mathfrak{h}$ consisting of elements of the form $e^{h}$, $h \in ...
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40 views

Subgroups of SO$(2)$

I'm doing an independent study with John Stillwell's Naive Lie Theory and I wanted to know if I'm on the right track. I'm just looking for some confirmation that these are acceptable answers. Find ...
2
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0answers
43 views

Linearly independent skew symmetric complex matrices having the least eigenvalues

Question: Let $A$, $B$ be two $5 \times 5$ (or $7 \times 7$) skew-symmetric complex matrices (i.e. $A^t = -A$), and suppose that $$ \forall t,s \in \mathbb{C}, \quad M(t,s):=(tA+sB)^*(tA+sB) \text{ ...
2
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1answer
43 views

Are bounded subsets of Lie groups totally bounded

Let $ G $ be a finite dimensional real Lie group, and take a bounded ball $ B_R(e) \subset G $ in it, coming from the Riemannian metric, which itself is induced from an inner product on $ \mathfrak{g} ...
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0answers
44 views

Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
0
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1answer
29 views

Induced actions.

Let $U$ be the subgroup of $GL_n$ consisting of all unipotent upper triangular matrices. Suppose that there is an action of $U$ on a variety $X$. Let $\mathbb{C}[U]$ be the group algebra of $U$ and ...
0
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1answer
22 views

Characters of a unipotent group.

Let $\def\C{\mathbb{C}}T = (\C^*)^n$. A character of $T$ is defined to be a homomorphism from $T$ to $\C^*$. The characters of $T$ is of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some ...
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0answers
29 views

Why is U(n) a real form of GL(n)

When $n=1$, we see that $U(1)$ is defined by the equation $z\bar z=1$, hence $a^2+b^2=1$ for $z=a+bi$. Taking complex $a,b$ we see that the solutions are nonzero complex points, hence $U(1)$ is ...