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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107 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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95 views

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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33 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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36 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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59 views

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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30 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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64 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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17 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
46 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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27 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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32 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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28 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
49 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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22 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 ...
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163 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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41 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
54 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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54 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 ...
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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
38 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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34 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
65 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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43 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
16 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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16 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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23 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 $$G = (G/G^o) \times \left( (G^o/R_u(G^o)) \ltimes R_u(G^o)\right)$$ ...
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1answer
34 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
40 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
44 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
votes
1answer
75 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 ...
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1answer
32 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 ...
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1answer
79 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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13 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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34 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
votes
1answer
42 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 ...
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1answer
39 views

Finite representations of the Euclidean Group

What are the finite dimensional indecomposable representations of the special Euclidean group in three-dimensions, SE(3)? To clarify, I'm asking about the group $$SE(3) = \left\{ \begin{pmatrix} ...
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38 views

Cartan's Criterion for Solvability

I'm trying to understand the proof of Cartan's Criterion for Solvability given here, and have two questions: On page 15, about half way down, we assert the following: If $\mathfrak{g}=\mathfrak{g}_0 ...
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22 views

How to write su(3) Lie algebra as a sum of two subspaces? [duplicate]

Let K,F⊂su(3) be subspaces, such that K⊕F=su(3), and K has a su(2) structure. How can we show that [K,K]=K (i.e., commutator of any two elements of K gives an element in K), [K,F]=F, and [F,F]=K?
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20 views

Adjoint representation for matrix groups (Gauge theory)

This is a question in regards to an identity in Gauge theory. Let $\omega$ be the connection one form on a principal bundle $\pi:P\rightarrow M$ and let $A_{\alpha}:=s_{\alpha}^*\omega$ be the gauge ...
4
votes
1answer
31 views

Sections of associated bundles

Let $\pi:P\rightarrow M$ be a Principal bundle and $\pi_V:P\times_G F\rightarrow M$ be its associated bundle via the representation $\rho:G\rightarrow GL(V)$. Fact: $\Gamma(P\times_G ...
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1answer
25 views

Connected Matrix Lie groups

I was reading Hall's book on Lie groups. After defining Connected Lie groups he stated and proved a proposition : If $G$ is a matrix Lie group then the component of $G$ containing identity is a ...
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20 views

Generators of the SU(2) matrix group

Let $X_i$ be the generators of $SU(2)$ and let the parameters of the rotation be $\theta, \phi, \delta$ such that the matrix $R = e^{i(\phi X_{1} + \delta X_{2} + \theta X_{3})}$, where $R$ is an ...
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58 views

Lie algebra of a lie group

In proposition 5.2 of Bump's Lie Groups, he states: Let $G$ be a closed Lie subgroup of $GL(n, \mathbb{C})$, and let $X \in Mat_n (\mathbb{C})$. Then the path $t \to exp(tX)$ is tangent to the ...
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1answer
33 views

The dimension of the SU(2) matrix group

Let's take the matrix $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Consider its transpose $R^\dagger = \begin{pmatrix} a^* & c^* \\ b^* & d^* \end{pmatrix}$. Then $RR^\dagger ...
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1answer
50 views

$\mathbb{R}^2$ as a quotient of a group

I am looking for a locally compact group $G$ with a closed subgroup $H$ such that $G/H$ is homeomorphic with $\mathbb{R}^2$ but $G$ does decompose into a semidirect and/or direct product which ...
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1answer
27 views

Dimensions of classical Lie groups

I understand that the dimension of the $SU(n)$ matrix group is $n^2$ because there are $2n^2$ real variables (for the $n^2$ complex matrix elements) in each matrix, and there are $n^2$ equations ...
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8 views

Question on Discrete series representations of semisiple Lie groups

I am reading Knapp's book, representation theory of semisiple Lie groups. I am confused with the statements in the following. In page 310, Theorem 9.20: Let $\lambda \in (i\mathfrak b)'$ is ...
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1answer
30 views

Property of left invariant vector field and its local flow.

Given $G$ a lie group and $X$ a left invariant vector field. Let $\Phi_X^t$ be the local flow of $X$. Why can we conclude that $\Phi_X^t \circ L_x=L_x \circ \Phi_X^t$? Thanks!
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1answer
135 views

Why are Lie algebras “rigid” objects?

I read the following motivation for quantum groups on wikipedia: The discovery of quantum groups was quite unexpected, since it was known for a long time that compact groups and semisimple Lie ...