# Tag Info

Accepted

### What is Lie Theory/ a Lie Group, simply?

This was going to be a comment, but it got too long. I don't know how "simple" this is, but the 5 word summary is "a group with manifold structure". Or perhaps if you're a ...
• 36.7k
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### Under what conditions is the exponential map on a Lie algebra injective?

There is a complete characterization, in a large part due to Dixmier and Saito (both independently in 1957): If $G$ is a real (finite-dimensional) Lie group with Lie algebra $\mathfrak{g}$, then ...
• 17.7k
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### Can every manifold be turned into a Lie group?

There is an easy counterexample: $S^2$ cannot be given a Lie group structure (this is a consequence of the hairy ball theorem). The problem with your construction is that it doesn't offer how to ...
• 2,067
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### When will two isomorphic Lie algebras have the same representation?

Short answer: If two Lie algebras are isomorphic, they have "the same" complex representations. A real semisimple Lie algebra and its complexification also have "the same" complex representations, but ...
• 25.3k
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### Do Lie algebras "know" about their Lie groups?

In principle it is possible to determine every (edit: connected) Lie group having Lie algebra some finite-dimensional real Lie algebra $\mathfrak{g}$. As radekzak says in the comments, first we find ...
• 415k
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• 97.9k

### Are all Lie groups Matrix Lie groups?

As other answers mention, it is not true that any Lie group is a matrix group; counterexamples include the universal cover of $SL_2(\mathbb{R})$ and the metaplectic group. However it is true that all ...
• 3,917
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### Can you give me an example of topological group which is not a Lie group.

Another example is the rationals under addition with the subspace topology induced from inclusion in $\mathbb R$. Since the rationals are countable, they can't be a manifold of dimension exceeding $0$,...

### Can every manifold be turned into a Lie group?

To add to the previous answers, topological groups have abelian fundamental groups. $G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian Orientable surfaces of genus at least two are not ...
• 1,437
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### Why is enveloping algebra called enveloping algebra?

1. The very first thing that is crucial to have in mind is that every associative algebra (where we have a vector space addition $+$, and a multiplication $\cdot$) can be turned into (or let's better ...
• 25.3k

### Quaternion–Spinor relationship?

Some background in Lie theory first. To any Lie group $G$ one may associate its lie algebra $\mathfrak{g}$. Among all isomorphism classes of Lie groups $G$ with a given lie algebra $\mathfrak{g}$, ...
• 151k
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### Are all Lie groups Matrix Lie groups?

Not all Lie groups are matrix groups. Consider the metaplectic group. From wikipedia: The metaplectic group $M_{p_2}(\mathbb{R})$ is not a matrix group: it has no faithful finite-dimensional ...
• 1,017

### Why do we care about two subgroups being conjugate?

$H_1$ and $H_2$ are conjugate as subgroups iff $G/H_1$ and $G/H_2$ are isomorphic as $G$-sets. Edit: Two related settings where this condition shows up are Galois theory and covering space theory. ...
• 415k

### Differential of the multiplication and inverse maps on a Lie group

One of Tu's previous exercises, ex. $8.7^*$ shows that for $M,N$ manifolds, by defining $\pi_1:M\times N\to M$ and $\pi_2: M\times N \to N$, one can show for $(p,q)\in M\times N$ that \pi_{1*}\times\...
• 4,305
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### Does a Lie group's group structure (not Lie group structure) determine its topology?

$\mathbb{R}^n$ and $\mathbb{R}^m$ are abstractly isomorphic (assuming the axiom of choice) for $n \neq m$ but not homeomorphic and so not isomorphic as topological groups. I think this might be the ...
• 415k

### prove $RP^3\cong SO(3)$

Each rotation in $\Bbb R^3$ is characterized by an "oriented axis" $v\in S^2$ and an angle $\varphi\in [0,\pi]$ and the only relations are $(v,\pi)=(-v,\pi)$ and $(v,0)=(w,0)$ for each $v,w\in S^2$. ...
• 11.4k

### Prove that the manifold $SO(n)$ is connected

We have that $S^n \cong SO(n+1)/SO(n)$. Using the fact that: If $H <G$ is a closed subgroup and both $H$ and $G/H$ are connected, then $G$ is connected, the claim follows by induction.
• 33.9k
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### Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

To see such relations more clearly, one should first of all avoid identifying the weight space $\Lambda\otimes\Bbb R$ with its dual space. Even in the semisimple case, an invariant inner product is ...
• 114k
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### Is there a way to associate a Lie algebra to the group of diffeomorphisms?

This question certainly is too broad to be answered completely. Before talking about infinite dimensional Lie groups you need a concept of infinite dimensional manifolds. As you note correctly, you ...
• 20k
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### Lie algebra: intuition of "Lie Algebra is tangent space of corresponding Lie Group"?

Intuitively, you can think of the tangent space of a surface at a point as the space of all "directions" you can move from that point (with all velocities) while staying on the surface. That ...
• 2,215
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### Name and layperson's explanation for an E8 group diagram.

The diagram you posted is not a Dynkin diagram, but is the projection of the convex hull of the root system $E_8$ in the Coxeter plane, i.e. the invariant plane for the action of the Coxeter element ...
• 5,041
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### Prove that the manifold $SO(n)$ is connected

Given two vectors $v,w$ of same length (say of length $1$) in $\mathbb{R}^n$, let's first verify that there is indeed a path in $\gamma(t) \in SO(n)$ so that $\gamma(t)v$ "starts off" at $v$ and "...
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### Why do we care about two subgroups being conjugate?

I will not attempt a complete answer, but just as an example, if you are studying finite simple groups $G$ (which lots of people do), then one of the first questions you might ask about $G$ is: what ...
• 87.8k
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### What is symplectic geometry?

Quick "fake" answer: In classical mechanics one usually describes a particle measuring its position $q_1, \dots, q_n$ and momentum $p_1, \dots, p_n$. To describe how these change one needs to ...
• 3,098
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### Example of a representation of $U(1)$ with $n>2$

Some remarks on language A representation of a group $G$ is indeed what you describe: a linear map $R$ from $G$ to a group $GL(n, \mathbb{C})$ of matrices. Most of the time, the vector space on which ...
• 10.3k