# Tagged Questions

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 (group-...

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### Help needed to understand statements about torus

I am having trouble understanding two statements: Let $A$ be an algebraic curve in $\mathbb{P}^2$ over $\mathbb{C}.$ Consider its normalization $$\pi: \hat{A} \to A.$$ If genus $g(\hat{A})=1,$ ...
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### lie group and lie algebra: confusion about the tangent space

I am reading a document about performing 2D and 3D transformations in space specifically rigid body transformations which can be represented using Lie groups. For example, the rigid body ...
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### Confused about classification of simple lie algebras

In the classification of simple lie algebra, I learnt that $\mathfrak{sl}(n + 1)$ has a root system $A_n$. For example http://stacky.net/files/written/LieGroups/LieGroups.pdf, page 82. But I found ...
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### Compact Lie group with non discrete center?

Could someone please give me an example of a compact Lie group with non discrete center which is not just a product of a group with a torus?
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### Tensor product over Lie group isomorphic to that over its Lie algebra

Hi: I have a question about the tensor product of representation of Lie groups as follows: Let $G$ be a connected, complex Lie group with Lie algebra $\mathfrak{g}$. Let $V$ and $W$ are ...
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### Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
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### 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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### Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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### Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ...
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### Periodic geodesics in special unitary groups

Looking at the solution of the equation of motion for the action using a bi-invariant metric on a Lie group. If I take the lie group as $SU(2)$ the geodesics are always periodic because $SU(2)$ is the ...
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### Why are root spaces of root decomposition of semisimple Lie algebra 1 dimensional?

I'm trying to understand root system of semisimple Lie algebra but having trouble following one of the step in the note which explain why each root spaces are 1-dimensional. According to the note, ...
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### Help understanding the argument used in this proof that $SO(n)$ is path-connected

Background: I am reading Tapp's matrix groups for undergraduates and I am in the process of showing that $SO(3)$ is path-connected. While working on my own arguemnt I found a proof online in these ...
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### Algebra of matrix coefficient over a compact group is isomorphic to its dual.

Let $K$ be a compact group. Then we have the following definition of matrix coefficient: Definition: $f: K \rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite dimensional ...
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### Need help with proof of $SO(3)$ is path connected

I am working on an exercise in Tapp's matrix groups for undergraduates. It is a proof that $SO(3)$ is path-connected. $SO(3)$ is the group $$SO(3) = \{A \in O(n)\mid \det A = 1 \}$$ where $O(n)$ is ...
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### Is $O(2)$ really not isomorphic to $SO(2)\times \{-1,1\}$?

An exercise in a book I'm reading is to show that $O(2)$ is not isomorphic to $SO(2)\times \{-1,1\}$. The problem is, I don't believe the statement. Let me elaborate why: $O(2)$ consists of ...
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### Isn't this $f$ always a group isomorphism

Consider the following exercise from a book I'm reading: If $n$ is odd show that $$f: O(n) \to SO(n) \times \{1,-1\}, A \mapsto (A \operatorname{det}{A}, \operatorname{det}{A})$$ is an ...
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### Product of two arbitrary elements in $O(2)$

An exercise in a book I'm reading asks to describe the product of two arbitrary elements in $O(2)$. I would like to solve the exercise but I got stuck. I know that an element in $SO(2)$ can be written ...
Suppose I have a Lie group action $$G\times M\to M, (g,m)\mapsto g\cdot m.$$ which is transitive on $M$, then the tangent functor $T$ induces a corresponding map: $$TG\times TM\to TM, (\delta g,\... 0answers 276 views ### What is the Jacobian for Sim(3) lie group action on 3D points ? (4d homogenous points) I am coding up Sim(3) constraint types for a factor graph, and need to calculate the jacobian of the Sim(3) group action on 3D points. I am following the guide on http://ethaneade.com/lie.pdf ... 0answers 17 views ### Topology of orthogonal groups when n > 4? On Wikipedia I read that the topologies of O(1) and SO(1) to SO(4) are known topological spaces. What about O(n), SO(n), U(n) when n>4? 1answer 298 views ### Proof help: SU(2) is a double cover of SO(3) I am reading a proof that SU(2) is a double cover of SO(3). My source is this set of notes: http://www.damtp.cam.ac.uk/user/examples/D18S.pdf. The proof begins near the bottom of page 4. I have ... 2answers 69 views ### How can I show that these matrices don't commute I want to show that A\in O(2) \setminus SO(2) and B \in SO(2) don't commute. To prove it I wrote$$ B = \left ( \begin{array}{cc} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \...
I am trying to show that $$F = \left(\begin{array}{cc} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array} \right )$$ is a flip about a line through the origin. What I ...