Questions tagged [lie-groups]

Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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148 votes
7 answers
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Homotopy groups of Lie groups

Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
Matt Noonan's user avatar
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141 votes
14 answers
47k views

Why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
Olivier Bégassat's user avatar
104 votes
3 answers
9k views

Has the Lie group E8 really been detected experimentally?

A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced, "Quantum ...
Richard Borcherds's user avatar
79 votes
26 answers
6k views

What would you want on a Lie theory cheat poster?

For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...
76 votes
7 answers
8k views

Example of a manifold which is not a homogeneous space of any Lie group

Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...
MTS's user avatar
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74 votes
7 answers
19k views

What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
Theo Johnson-Freyd's user avatar
72 votes
6 answers
7k views

Surprisingly short or elegant proofs using Lie theory

Today, I was listening to someone give an exhausting proof of the fundamental theorem of algebra when I recalled that there was a short proof using Lie theory: A finite extension $K$ of $\mathbb{C}$...
64 votes
6 answers
8k views

Origin of terms "flag", "flag manifold", "flag variety"?

These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably ...
Jim Humphreys's user avatar
59 votes
8 answers
12k views

Why the Killing form?

I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
Ryan Reich's user avatar
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56 votes
4 answers
13k views

Group theory in machine learning

I'm a Machine Learning researcher who would like to research applications of group theory in ML. There is a term "Partially Observed Groups" in machine learning theory which has been ...
drosophyllum's user avatar
52 votes
5 answers
7k views

Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
zroslav's user avatar
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52 votes
2 answers
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Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?

$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
Eugene Stern's user avatar
51 votes
2 answers
2k views

$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was lead to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free. ...
André Henriques's user avatar
47 votes
7 answers
14k views

Classification of (compact) Lie groups

I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your ...
41 votes
9 answers
6k views

Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?

Harold Williams, Pablo Solis, and I were chatting and the following question came up. In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
Anton Geraschenko's user avatar
41 votes
3 answers
3k views

What is the classifying space of "G-bundles with connections"

Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a $...
Theo Johnson-Freyd's user avatar
39 votes
3 answers
4k views

Can every Lie group be realized as the full isometry group of a Riemannian manifold?

Suppose a finite-dimensional Lie group $G$ is given. Does there exist a connected manifold $M$ and a Riemannian metric $g$, such that $G$ is the full isometry group of $(M,g)$? For example if I try to ...
Panagiotis Konstantis's user avatar
38 votes
18 answers
23k views

Learning about Lie groups

Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
Daniel Erman's user avatar
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36 votes
6 answers
4k views

Examples of applications of the Borel-Weil-Bott theorem?

In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes: A representation Ri of a group G should be seen as a quantum object. This ...
Theo Johnson-Freyd's user avatar
35 votes
5 answers
4k views

$G_2$ and Geometry

In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie ...
Sean Tilson's user avatar
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35 votes
3 answers
1k views

Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

We fix $G=\mathrm{SL}_3(\mathbf{R})$. Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$? Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
YCor's user avatar
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34 votes
6 answers
6k views

Is SO(4) a subgroup of SU(3)?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$I want to write a $3 \times 3$ complex-matrix representation of $\SO(4)$, for example, we know that $\SO(5)$ is a subgroup of $\SU(4)$, so we ...
Faraz Mehdi's user avatar
34 votes
1 answer
1k views

Is there a geometric construction of hyperbolic Kac-Moody groups?

Just as the theory of finite-dimensional simple Lie algebras is connected to differential geometry and physics via the theory of simple Lie groups, the theory of affine Lie algebras was connected to ...
John Baez's user avatar
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33 votes
1 answer
4k views

Isometry group of a homogeneous space

Background Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ ...
José Figueroa-O'Farrill's user avatar
33 votes
3 answers
6k views

When is a finite dimensional real or complex Lie Group not a matrix group

I have a smattering of knowledge and disconnected facts about this question, so I would like to clarify the following discussion, and I also seek references and citations supporting this knowledge. ...
Selene Routley's user avatar
32 votes
1 answer
2k views

Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$

Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
John Pardon's user avatar
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31 votes
8 answers
8k views

"Modern" proof for the Baker-Campbell-Hausdorff formula

Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula? All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and are not at all geometric (...
Mark.Neuhaus's user avatar
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31 votes
2 answers
3k views

In any Lie group with finitely many connected components, does there exist a finite subgroup which meets every component?

This question concerns a statement in a short paper by S. P. Wang titled “A note on free subgroups in linear groups" from 1981. The main result of this paper is the following theorem. Theorem (Wang, ...
Khalid Bou-Rabee's user avatar
31 votes
2 answers
1k views

Is Lie group cohomology determined by restriction to finite subgroups?

Consider the restriction of the group cohomology $H^*(BG,\mathbb{Z})$, where $G$ is a compact Lie group and $BG$ is its classifying space, to finite subgroups $F \le G$. If we consider the product of ...
overcaffeinated's user avatar
31 votes
3 answers
2k views

What's the supersymmetric analogue of the Monster group?

Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...
John Baez's user avatar
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30 votes
7 answers
7k views

Why is Lie's Third Theorem difficult?

Recall the following classical theorem of Cartan (!): Theorem (Lie III): Any finite-dimensional Lie algebra over $\mathbb R$ is the Lie algebra of some analytic Lie group. Similarly, one can propose ...
Theo Johnson-Freyd's user avatar
29 votes
3 answers
3k views

The non-simplicity of $SO(4)$ and $A_4$

It is well known that the alternating group $A_n$ is simple unless $n=4$. It is likewise well known that the special orthogonal group $SO(n)$ is essentially simple unless $n=4$ (specifically, the ...
Drew Armstrong's user avatar
29 votes
2 answers
767 views

Does $\mathrm{SO}(3)$ act faithfully on a countable set?

Let $\mathrm{SO}(3)$ be the group of rotations of $\mathbb{R}^3$ and let $S_\infty$ be the group of all permutations of $\mathbb{N}$. Is $\mathrm{SO}(3)$ isomorphic to a subgroup of $S_\infty$? This ...
Erik Walsberg's user avatar
29 votes
2 answers
1k views

Gaussian elimination is just Gram-Schmidt with a change to the inner product symbol?

I noticed at some point that if you take the Gram-Schmidt algorithm for taking the QR decomposition of a matrix, and you change the meaning of the inner product symbol $\langle \mathbf u, \mathbf v \...
wlad's user avatar
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29 votes
2 answers
3k views

Simple discrete subgroups of Lie groups

Upon Ian Agol's suggestion, I separated this question from the one on non-residual finiteness in Non-residually finite matrix groups Question. Are there infinitely generated simple discrete ...
Misha's user avatar
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28 votes
3 answers
3k views

Rep Theory Consequences of Bott--Weil--Borel

I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...
Jean Delinez's user avatar
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28 votes
2 answers
2k views

What is special to dimension 8?

Dimension $8$ seems special, as the partial list below might indicate. Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$? Surely it isn't it, in the end, simply because ...
27 votes
5 answers
3k views

Is there a Morse theory proof of the Bruhat decomposition?

Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...
skupers's user avatar
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27 votes
10 answers
2k views

Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. What are some good examples of Kan extensions, adjunctions, and (co)...
27 votes
4 answers
4k views

Triality of Spin(8)

Among simple Lie groups, $Spin(8)$ is the most symmetrical one in the sense that $Out(Spin(8))$ is the largest possible group. A description of this outer automorphism groups is as follows. $Spin(8)$ ...
Aliakbar Daemi's user avatar
27 votes
3 answers
3k views

Is there a 'nice' interpretation of virtual representations?

Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...
ARupinski's user avatar
  • 5,181
27 votes
2 answers
1k views

Intuition for symplectic groups

My question essentially breaks down to How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
Robin Goodfellow's user avatar
27 votes
1 answer
848 views

Why do the adjoint representations of three exceptional groups have the same first eight moments?

For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...
Will Sawin's user avatar
  • 135k
27 votes
1 answer
3k views

Definitions of real reductive groups

There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind: A closed subgroup of $GL(n,\mathbb C)$ closed under conjugate transpose. The set ...
AndreA's user avatar
  • 971
26 votes
5 answers
9k views

Textbook or lecture notes in topological K-Theory

I am looking for a good introductory level textbook (or set of lecture notes) on classical topological K-Theory that would be suitable for a one-semester graduate course. Ideally, it would require ...
26 votes
2 answers
4k views

Cohomology of Lie groups and Lie algebras

The length of this question has got a little bit out of hand. I apologize. Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...
algori's user avatar
  • 23.2k
25 votes
1 answer
892 views

Are (semi)simple Lie groups some sort of "homotopy quotient groups" of their maximal tori?

Warning: non-specialist writing, some rubbish possible. The formula $h^*(BG)\cong h^*(BT)^W$ valid for complex oriented cohomology of the classifying space of a compact Lie group $G$ with maximal ...
მამუკა ჯიბლაძე's user avatar
25 votes
1 answer
691 views

The de Rham complex of the octonionic projective spaces

The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...
Nadia SUSY's user avatar
24 votes
2 answers
2k views

Is it possible to realize the Moebius strip as a linear group orbit?

On MSE this got 5 upvotes but no answers not even a comment so I figured it was time to cross-post it on MO: Is the Moebius strip a linear group orbit? In other words: Does there exists a Lie group $ ...
Ian Gershon Teixeira's user avatar
24 votes
3 answers
2k views

Spin group as an automorphism group

Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with $p$...
Mikhail Borovoi's user avatar

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