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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103 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 ...
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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 ...
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46 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 ...
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122 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. ...
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1answer
89 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 ...
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104 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 ...
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134 views

Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?
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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 ...
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1answer
95 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 ...
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215 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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1answer
37 views

Group of isometries is closed in $GL_{n+1}$

I solved the following exercise. Could someone please check my work? Prove that $\operatorname{Isom_n}{\mathbb R^n}$ is a matrix group. Is it compact? My work: We recall that ...
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41 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) ...
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1answer
62 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
42 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 ...
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1answer
144 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 ...
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1answer
123 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 ...
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1answer
437 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) ...
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29 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 $$ ...
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2answers
80 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.
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1answer
91 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 ...
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70 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 ...
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1answer
39 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 ...
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179 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 ...
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144 views

Maximal tori in U(2)

I'm attending a course about Lie Groups, and in an exercise I'm asked to "find the maximal abelian subgroup of $U(2)$". Certainly an abelian subgroup of $U(2)$ (and in fact of any $U(n)$ increasing ...