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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Weights in the Dynkin Basis and Eigenvalues of the Cartan Generators for SU(3)?

The Cartan Generators of $SU(3)$ in the three dimensional rep have eigenvalues $(1,-1,0)$ and $\frac{1}{\sqrt{3}} (1,1,-2)$. Therefore we have the weights: $$ (1,\frac{1}{\sqrt{3}}) \quad ...
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1answer
21 views

Connected matrix Lie group

While enjoying Lie groups with Brian C. Hall's "Lie groups, Lie algebras, and representations", I'm stuck with the "standard argument using the compactness of the interval $[0,1]$" in the proof of the ...
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34 views

Irreducible unitary representations of $ \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $.

Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ ...
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19 views

Irreducible unitary representation of a solvable lie group

Determine the equivalence classes of irreducible unitary representations of a solvable lie group. $$\begin{bmatrix}ae^t & 0\\ 0 & be^{-t}\end{bmatrix}$$ for $a,b\in \mathbb{R}, t\in ...
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11answers
260 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$?

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
3
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19 views

homomorphism between smooth algebraic groups of the same dimension

For Lie groups, we have a theorem: Suppose $G$ and $G'$ are Lie groups of the same dimension, $G'$ is connected, and $f : G \to G'$ is a homomorphism of Lie groups with discrete kernel. Then, $f$ ...
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36 views

Cartan-Maurer trivializations and Bi-invariant Metric on Lie Group

Let $G$ be a Lie group. Let $\nabla^L$ and $\nabla^R$ be the connections on $TG$ corresponding to the trivial connection $d$ on $G\times\mathfrak{g}$ under the left and right trivializations. How ...
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1answer
17 views

What really is a path-ordered exponential?

In some texts about gauge theories in Physics I've found one object called a path-ordered exponential which I'm not sure what it means. As I understood, the idea is as follows: let $G$ be a Lie group ...
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26 views

Lie algebras and left or right invariant vector fields

Let $G$ be a Lie group and $\phi$ be a diffeomorphism defined by $\phi(\sigma)=\sigma^{-1} $, $\sigma\in G$. I have to prove that $X\mapsto d\phi(X)$ gives an isomorphism of algebras between the ...
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1answer
28 views

why is $S^2$ not a Lie group?

I'm reading John Stillwell's "Naive Lie Theory" and it was mentioned there (without giving a proper definition of what a Lie group is) that the only Lie groups among the unit n-spheres are $S^1$ and ...
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1answer
36 views
+150

Explicit homotopy equivalence of homogeneous spaces $O(2n)/U(n)$ and $GL(2n,\mathbb{R})/GL(n,\mathbb{C})$

Exercise 2.25 of symplectic topology by McDuff and Salamon asks me to prove that $O(2n)/U(n)$ is homotopy equivalent to $GL(2n,\mathbb{R})/GL(n,\mathbb{C})$. They suggest to use the polar ...
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94 views
+50

$5$ dimensional space over $\mathbb{R}$

When coming up with a double cover of $SO(5)$, I used conjugation by matrices of the form $$\begin{pmatrix} r & q\\ \overline{q} & r \end{pmatrix}$$ where $r\in\mathbb{R}$ and $q$ is a ...
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1answer
24 views

How to show that for every element $g$ in a Lie group, the curve $\gamma(t) = g\gamma_v(t)$ is an integral curve such that $\gamma(0) = g$?

Could anyone help me a little with how to begin to solve the following problem? Thanks in advance! Definitions Let $G$ be a Lie group. For $g \in G$, define a diffeomorphism $l_g$ of $G$ by ...
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1answer
52 views

The Plate Trick and $SO(3)$

This is a followup of sorts to Plate trick demonstrating SO(3) not simply connected. . Suppose I do the ``plate trick'' to demonstrate the existence of an order-2 element in $\pi_1(SO(3))$. That is, ...
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1answer
97 views

How could a group be a manifold?

For example a Lie group is defined as a certain differentiable manifold, but what does this mean geometrically, and what is gained by viewing something abstract and algebraic as a manifold? First, I ...
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68 views
+50

Right invariance of Casimir (Laplacian)

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. The Casimir element $\Delta\in \mathfrak{zu}(\mathfrak{g})$, considered as an operator on $C^{\infty}(G)$ is right invariant, that is, $\Delta ...
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1answer
29 views

Are the torsion elements dense in every compact Lie group?

Let $ G $ be a compact connected real Lie group. Denote by $ T $ its set of torsion elements. Is $ T $ always dense in $ G $?
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30 views

Can the following simple tensor preserving map exist?

In this question here, I asked if there could exist a $U \in U(4)$ such that $U$ itself was not the tensor product of two matrices, but such that $U(A \otimes B)U^{-1} = A' \otimes B'$ for all $A,B ...
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42 views
+50

At what position do we insert the new coefficient in the weights for extended Dynkin Diagrams?

Given a set of weights of a representation and the corresponding extended Dynkin diagram for some Lie algebra, we can delete a node, which yields the maximal subalgebra. I know how to draw the ...
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1answer
20 views

Adjoint representation and tangent vectors

Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra, $\text{Ad}:G\rightarrow GL(\mathfrak{g})$ the adjoint representation of $G$. Then, for $X,Y\in \mathfrak{g}$, \begin{align*} ...
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53 views

Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
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0answers
16 views

What is an infinitesimal automorphism of a connection?

Let $P\to M$ be a principal bundle over a differentiable manifold $M$, with fibre $G$, equipped with a connection $H\subset TP$. I have heard the term "infinitesimal automorphism of $H$" but I haven't ...
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1answer
40 views

Why can we write the weights of a representation in terms of the simple roots?

I'm currently trying to get my head around the fact that we can write the weights of any representation in terms of the simple roots of the algebra. Is there any, not too-technical, explanation? I ...
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26 views

How can I find the Weights of a Subalgebra

I'm currently trying to understand how we can derive the weights of a subalgebra of a given representation of a Lie group. For example, if we start with the 16-dimensional representation of ...
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0answers
25 views

How to prove the Lie bracket is infinitesimal commutator

I am currently studying Lie groups and I cannot solve the following exercise, which I think is vital to my understanding. The Lie bracket is defined as $[X,Y]=\text{ad}(X)Y$. Let the group commutator ...
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0answers
26 views

Sp(2n) as manifold

How to prove that $Sp(2n)$ is a manifold? We know that $Sp(2n)\subset Gl(2n)$ and $Gl(2n)$ is a manifold. Furthermore $Sp(2n)$ can be described as zeros of $A\mapsto A^TJA-J $, where $J$ is a ...
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1answer
37 views

Killing forms and Hermitian inner products

Let $K$ be a compact, connected, simply connected Lie group with Lie algebra $\mathfrak k$ and Killing from $B_{\mathfrak k}$. It is well known that $B_{\mathfrak k}$ is a negative definite symmetric ...
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20 views

Subspaces of Lie algebras

The Lie correspondence is well understood. For 'nice enough' Lie groups $G$ (with Lie algebra $\mathfrak{g}$) every sub-group $H < G$ has a Lie algebra $\mathfrak{h} < \mathfrak{g}$ given by ...
12
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1answer
143 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose ...
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38 views

Is the Groupoid of germs associated to an orbifold a Hausdorff proper Lie groupoid?

I was studying the book Introduction to Foliations and Lie Groupoids by I. Moerdijk and J. Mrcun and I have a doubt. On page 140 they give the following definition for a proper Lie groupoid. A ...
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1answer
36 views

Fibre bundles for homogeneous spaces

Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then it is known that $G/H$ can be equipped with a unique differentiable structure such that $G\xrightarrow{\pi} G/H$ is a locally ...
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1answer
45 views

How can I maintain linear independence through a commutator?

Consider a Lie algebra $\mathcal{L}$, a linearly independent generating set $\mathcal{G}$, and an element $X \in \mathcal{L}$. Edit: Note that $\mathcal{G}$ is not necessarily a basis; the generation ...
3
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1answer
27 views

Vector space operations on fibres of associated bundles.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\text{ad}:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let ...
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1answer
33 views

Am I correct in saying that there are no non-commuting connected one-parameter Lie groups?

I posed myself a question a while ago: are there any non-commuting one parameter Lie groups? I'm thinking that there are no non-commuting connected Lie groups (not sure how to proceed to the ...
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0answers
33 views

Can the adjoint representation on a discrete centered group merge connected components?

Let $\mathfrak{G}$ be a Lie group with algebra $\mathfrak{g}$ and with in general many connected components but which is either centerless or has at most a discrete center. Then we have an ...
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1answer
64 views

If $ g \circ f$ is real analytic and $g$ is a real analytic immersion, then $f$ is real analytic

Let $M$ $N$ $P$ be complex manifolds, and let $$f: M\rightarrow N, g: N\rightarrow P$$ be $C^\infty$ maps with $g$ and $g\circ f$ holomorphic, and with $dg$ never degenerate. It's easy, then, to see ...
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2answers
37 views

How to descent to smaller groups “by chopping off a node of the Dynkin diagram”?

I read in section 2 of this paper : "There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram." What exactly is here referring to ...
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1answer
13 views

Why is the rank of a group is equivalent to the maximum number of independent U(1) generators?

I read here http://motls.blogspot.de/2012/04/exceptional-lie-groups.html that the rank of group is "the maximum number of independent U(1) generators". In my understanding the rank of a group ...
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0answers
14 views

Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
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17 views

Decomposition of an algebraic group into unipotent radical, component group and Levi subgroup

Consider the following statement : Given an algebraic group $G$ (over $\mathbb{R}$ or $\mathbb{C}$), there is a decomposition $$G = (G/G^o) \times \left( (G^o/R_u(G^o)) \ltimes R_u(G^o)\right)$$ ...
3
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1answer
30 views

Why not every homogeneous manifold is parallelizable?

It is obvious that not every homogeneous manifold is parallelizable (take for example the two-sphere $S^{2}$). In contrast, every Lie group $G$ is parallelizable, as you can construct a pointwise ...
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1answer
27 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can ...
5
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1answer
35 views

Representing Matrix Subgroups

Suppose I wanted to describe the subgroup of $GL_n(\mathbb{R})$ of matrices of the form $$ \left [ \begin{array}{cc} A & B \\ 0 & C \\ \end{array} \right ] $$ where $A \in ...
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19 views

Is trace of regular representation in Lie group a delta function? [duplicate]

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
2
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1answer
64 views

Is trace of regular representation in Lie group a delta function?

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
1
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1answer
24 views

Gauge transformation on a principal bundle

I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by ...
1
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1answer
60 views

direct sum and tensor product of representation of lie algebra

Let $(p_1,V_1)$ , $(p_2,V_2)$ representation of a lie algebra $g$ on $V_1,V_2$. I have to prove that: $ i) $ the direct sum $p_1 \oplus p_2$ is a representation of $g$ in $V_1 \oplus V_2$ $ ii) $ ...
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0answers
10 views

Commutation Relation - Generators of Semisimple Lie Group

The following is stated in a book I picked up, Group Theory for High-Energy Physicists - M. Saleem, M. Rafique. Consider an $r$-parameter semisimple Lie group of rank $\ell$. It has a set of ...
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33 views

Is there a general method to calculate the generators of the subgroups of $\textrm{GL}(n,F)$?

I know this might be a very bad/broad question, but after going through a few practice problems for finding linearly independent generators for some of the easier subgroups of $\textrm{GL}(n,F)$ ...
2
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1answer
39 views

What it means SO* (2N)?

I'm puzzled about the $"*"$ in the following notation for Lie groups: $SO^* (2N)$ or $SU^* (2N)$. I don't understand what is the meaning of this notation. It is introduced for example in Gilmore ...