# Tagged Questions

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### Why is $\mathfrak{sl}(n)$ the algebra of traceless matrices?

I'm studying Lie algebras as purely algebraic objects, without much of a background in the differential geometry surrounding Lie groups. The definition of $\mathfrak{sl}(n)$ has been given to me as ...
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### What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
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### Prerequisites to “Applications of Lie Groups to Differential Equations”

I'm currently a 4th year student at a university. I've become close with a professor and we talked about the topic of lie groups in differential equations. He then offered to do a reading course with ...
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### The Weyl group of A_3

Could someone please list all elements of the Weyl group of the root system $A_3$ in terms of simple reflections. In this case the Weyl group is $S_4$. Its strange that GAP failed to list all elements ...
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### Question on unitary representation of non-compact simple Lie groups

The following is an exercise appearing page 148 in Knapp's book, representation theory of semisimple groups. Let $G$ be a connected linear non-compact Lie group with simple Lie algebra $\mathfrak g$. ...
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### Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
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I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
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### What are the root systems for the n-dimensional torus?

My question may seem silly at first, but currently I am not able to work out the question of finding all roots for the n-dimensional torus. At first, it seemed obvious to me that there are no roots at ...
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### Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
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### Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
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### Lie group and generated ideals

I have this question in my textbook, and I can't seem to solve it on my own: Let $P \subset GL(n,\mathbb{C})$ be a subgroup as following: $P$ consists of all matrices in block ...
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### Does “Spherical Symmetry” as defined in General Relativity imply a Foliation of Spheres?

In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the ...
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### Lie groups. How to show that the group operations are smooth.

$N:=\{g\in GL(n,R) : g_{ij}=0 \forall j>i , g_{ii}=1 ∀i\}$. For this matrix group, how can we show that it is a Lie group? I am at the beginning of the subject of Lie groups so I can not ...
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### Is $sp(4)$ a subalgebra of $su(5)$?

Is $sp(4)$ a subalgebra of $su(5)$? And how can I prove/disprove this? I know already that it cannot be a regular maximal subgroup of $su(5)$ since the Dynkin diagram (which has two roots of unequal ...
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### Lie groups, maps and the Weyl group

If I have a map of simple Lie groups $H \to G$, do I get a map of Weyl groups $W_H \to W_G$? If $H$ is the semisimple component of a parabolic subgroup then we can clearly get this (see Ivan's answer ...
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### A maximal subalgebra of $E_6$ !?

I'm puzzeled by the following sentence in one of Baez's posts: The Lie algebra $E_6$ has a subalgebra of maximal rank isomorphic to $\mathfrak{so}(10)\oplus \mathfrak{u}(1)$. However, I thought ...
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### The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
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### Where does the name “toral” come from?

Where does the name "toral" come from in "toral subalgebra"? I know a little (very little) Lie groups theory, so I guess it could be related to a Lie group whose Lie algebra is the toral one. Is ...
I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...