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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385 views

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
3
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2answers
124 views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
3
votes
3answers
138 views

Differential of the inversion of Lie group [duplicate]

Let $G$ be a Lie group and $\iota \colon G\to G$ denote the inversion. If $e$ is the identity of $G$, prove that: $$\text d \iota _e = -\text {id} _{T_e G}.$$ I understand that the differential at $e$ ...
3
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1answer
88 views

Is there a Peter-Weyl theorem for the quasi-invariant measure on a homogeneous space of a compact semisimple group?

Let $H \hookrightarrow G$ be an inclusion of semisimple, compact Lie groups. There is a measure on the homogeneous coset space $G/H$ by pulling back the Haar measure on $G$ via the projection $G ...
3
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1answer
113 views

Subgroups of $SO(4)$ with free transitive action on $S^3$

By considering $S^3$ as the group manifold of $SU(2)$, the ordinary action of $SO(4)$ on the three sphere can be written as the $SU(2)\times SU(2)/\mathbb{Z}_2$ given by the group action of ...
3
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1answer
46 views

Commutativity and Maximal Tori in Connected, Compact Lie Groups

Let $G$ be a path-connected, compact Lie Group. Let $x \in G$ and let $T_x \subset G$ denote the union of all the maximal tori in $G$ that contain $x$. Question: Is it true that if $y \notin T_x$, ...
3
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1answer
104 views

Contractibility vs. G-contractibility

Let $X$ be a space equipped with an action of a compact Lie group $G$. Recall that such a space is said to be $G$-contractible if the identity map of $X$ is $G$-homotopic (i.e., homotopic through ...
3
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2answers
399 views

Local Isomorphism on Topological Groups

I'm currently studying Lie Groups by "Theory of Lie Groups I", C. Chevalley. He talks about Topological Groups on chapter two. To be more precise, on page 38 he presents two examples in order to show ...
3
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1answer
166 views

How to write down the pull back of a differential form by exponential map?

The exponential map $e_{m}: M(n,\mathbb{R})\rightarrow M(n,\mathbb{R})$ is defined by $$e_m(\alpha)=me^\alpha,\quad e^\alpha=1+\alpha+\frac{\alpha^2}{2}+\frac{\alpha^3}{3!}+\cdots$$ Now fix $q\in ...
3
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1answer
72 views

Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$

Let $G=GL_n(\mathbb{C})$. Scenerio 1: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ g.(x,y)=(gx,y). $$ Scenerio 2: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ ...
3
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1answer
481 views

Derivative of function including matrix logarithm

Is the following equation a first order approximation or incorrect for general matrix Lie groups? And what are the higher order terms? $$\frac{\partial}{\partial\mathbf x} (\log(\mathtt ...
2
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1answer
37 views

Lie Subgroup Example - Explanation?

I'm currently working through Jeff Lee's 'Manifolds and Differential Geometry'. He defines a Lie Subgroup, $H$, to be an abstract subgroup of a Lie Group $G$, such that the inclusion map ...
2
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1answer
94 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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0answers
113 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 ...
2
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1answer
29 views

$Hom_G(\pi,\sigma)$ = $Hom_{\mathfrak{g}}(d\pi,d\sigma)$?

Let $G$ be a Lie group. Let $\mathfrak{g}$ be the corresponding Lie algebra. Let $(\pi,V)$ and $(\sigma, W)$ be representations of $G$, with corresponding differentials $d\pi$ and $d\sigma$, which are ...
2
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0answers
49 views

Relationship between O(n)- and SO(n)-representations?

Write $O(n)$ and $SO(n)$ for the orthogonal and special orthogonal group of degree $n$ over the real numbers. Suppose that $V$ and $W$ are real, finite-dimensional and orthogonal ...
2
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2answers
128 views

Unitary group and unit circle

Let $U(n)$ denote the group of complex unitary matrices, let $S^1$ be the unit circle in the complex plane. Then the map $$f:U(n)\to S^1,\quad f(A)=det(A)$$ is a group homomorphism and a submersion. ...
2
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1answer
114 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
2
votes
2answers
91 views

Computing the differential of the map $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$

Let $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ be the map $$(B_1,B_2)\mapsto [B_1,B_2]$$ which takes two $2\times 2$ matrices to its Lie bracket. Then why does ...
2
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1answer
96 views

$SL(3,\mathbb{C})$ acting on Complex Polynomials of $3$ variables of degree $2$

So I'm given the following definition: $h(g)p(z)=p(g^{-1}z)$ where g is an element of $SL(3,\mathbb{C})$, $p$ is in the vector space of homogenous complex polynomials of $3$ variables and $z$ is in ...
2
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0answers
219 views

What is the constant $e$, fundamentally? [duplicate]

Possible Duplicate: Why is the number e so important in mathematics? Intuitive Understanding of the constant “e” The number $e$ is important in many respects. If you ask ...
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0answers
22 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
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2answers
63 views

Do these vector fields span an integrable distribution?

Let $X_i$, $1 \le 1 \le n$, be smooth vector fields on the open set $U \subset \mathbb{R}^n$, which are linearly independent at each point. Set $[X_i, X_j] = \sum_{k=1}^n c_{ij}^kX_k$. Suppose that ...
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0answers
42 views

How do non-semisimple modules over $\mathbb C\mathrm{GL}_n(\mathbb C)$ look like?

[Separated from another question] Can you give an example of a non-semisimple module over $\mathbb C\mathrm{GL}_n(\mathbb C)$? (Preferably one without direct summands, i.e. an indecomposable module) ...
1
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1answer
65 views

Is it true for solving differential equations by getting constant coefficient matrix with magnus expansion

The magnus expansion is given in detail http://en.wikipedia.org/wiki/Magnus_expansion. While implementing magnus expansion to differential equations we have an iteration formula as follows $$Y'(t) = ...
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1answer
44 views

What 1D $\mathbb{C}$-Subspaces are Stabilized by Elements of a Specific 2-Torus in $SO(7)$?

Consider the 2-torus $T \subset SO(7)$ defined by $T = \left\{ \mathrm{diag}(R_{\theta_1}, R_{\theta_2}, R_{-(\theta_1 + \theta_2)}, 1) \mid \theta_1, \theta_2 \in \mathbb{R} \right\}$, where ...
1
vote
1answer
148 views

Sp(1) and SO(3) Locally Isomorphic - Trouble with part of proof

My apologies if this is fairly basic. I'm trying to understand the proof that $Sp(1)$ and $SO(3)$ are locally isomorphic but are not isomorphic but have run into some trouble. Here's what my ...
1
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1answer
190 views

Dimension of Lie algebra according to root system

I was wondering how is it possible to find the dimension of a semi-simple lie algebra $L$ if its corresponding root system is (lets make it simple) of type $B_2$. We can find the number of roots and ...
1
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1answer
128 views

Weyl character formula and finding the trace.

Let $v$ be a positive integer. I have a representation $\rho_v$ of $USp(4) = \{g\in M_2(\mathbb{H})\,|\,g^{T}\bar{g}\}$, where $\mathbb{H}$ is Hamiltons quaternions. The representation $\rho_v$ has ...
1
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1answer
445 views

Representations of U(n) using bosons and fermions

I would be glad if someone can help me understand the argument in the first paragraph of page 4 of this paper. Especially I don't understand their first sentence, "Using N bosons (fermions) ...
0
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0answers
31 views

Could someone check my work on why $\mathrm{Aff}$ is a matrix group

I solved the following exercise, please could someone check my work? Prove that $\operatorname{Aff_n}{(\mathbb K)} \subseteq GL_{n+1}(\mathbb K)$ is a matrix group where $$ ...
0
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2answers
84 views

SO(n) is parallelizable

Prove that $SO(n)$ is parallelizable. How would I go about showing this? My supervisor could not help me with this problem, and I am stumped.
0
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1answer
106 views

In Lie groups, why is left translation a diffeomorphism?

$G \times G \rightarrow G$ defined as $(x,y) \rightarrow xy$ is differentiable. The left translation is $L_x(y) = xy$. To show that it is a diffeomorphism, we need it to be a bijection, differentiable ...
0
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0answers
76 views

Closed Lie subgroups of $SU(3)$

I'm looking for a reference describing the closed connected Lie subgroups of $SU(3)$. I know they are $SU(2)\times U(1)$, $SU(2)$, $SO(3)$ and several abelian subgroups based on this mathoverflow ...
0
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1answer
40 views

Does $\mathrm{Im}(\exp)$ being a manifold imply the domain is a manifold?

Let $G$ be a real matrix group of dimension $n$. Let $\mathfrak{g}$ denote the lie algebra of $G$. Suppose $X \subset \mathfrak{g}$ such that $e^X$ is a submanifold of $G$ of dimension $m$. Does ...