**57**

votes

**7**answers

7k views

### 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, ...

**19**

votes

**17**answers

11k 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.

**21**

votes

**6**answers

5k 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 ...

**42**

votes

**7**answers

7k 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 ...

**32**

votes

**7**answers

4k 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 ...

**9**

votes

**3**answers

1k views

### Definitions of Reductive and Semisimple Groups

I'm a graduate student. I've been reading Knapp's two books Representation Theory of Semisimple Groups and Lie Groups Beyond an Introduction. He seems to give wildly different definitions for the ...

**17**

votes

**1**answer

1k views

### What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...

**17**

votes

**2**answers

1k views

### Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?

Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a
complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ...

**4**

votes

**1**answer

256 views

### The existence of a finite dimensional Lie algebra with a given symmetric invariant metric

The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...

**4**

votes

**1**answer

397 views

### Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...

**10**

votes

**3**answers

448 views

### Nearby homomorphisms from compact Lie groups are conjugate

I'm looking for a proof (that I can understand) of the following fact: If $K$ and $G$ are Lie groups, and $K$ is compact, then nearby homomorphisms $K\to G$ are conjugate.
That is, if ...

**8**

votes

**3**answers

719 views

### subgroup of SU(N) with maximal manifold dimension

Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not ...

**7**

votes

**1**answer

814 views

### unitary irreps of O(p,q)

I am interested in the irreducible unitary representations of the orthogonal groups $O(p,q)$. By $O(p,q)$ I mean the real Lie groups which preserve the quadratic form of signature $(p,q)$ in ...

**3**

votes

**2**answers

467 views

### Complete classification of six dimensional non-semi simple Lie algebra

I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$).
In this paper, in page 7, it ...

**2**

votes

**3**answers

449 views

### Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds

I would like to find a pedagogical reference where the classification, up to isomorphism, of principal $SU(2)$ bundles over a four-dimensional compact, oriented manifold is explained. In particular I ...

**4**

votes

**3**answers

249 views

### Generalization of a theorem of Burnside to non-compact groups

The following two theorems are often attributed to Burnside:
Theorem Let $G$ and $H$ be compact groups. Then the irreducible representations of $G\times H$ are precisely the representations ...

**3**

votes

**0**answers

582 views

### Has the Lie group preserving a probability distribution been used in Bayesian statistics?

For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes ...

**2**

votes

**1**answer

130 views

### Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's “Classical Groups”

First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but ...

**64**

votes

**11**answers

16k 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 ...

**83**

votes

**3**answers

6k 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 ...

**31**

votes

**3**answers

1k 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 ...

**18**

votes

**1**answer

2k 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 $M$ ...

**11**

votes

**3**answers

1k views

### Representations of \pi_1, G-bundles, Classifying Spaces

This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first ...

**22**

votes

**2**answers

3k 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 ...

**9**

votes

**6**answers

2k views

### cohomology of BG, G compact Lie group

It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group
$G$. However, I've still not found a proof of this. I believe that the proof is as follows:
--> $G$ compact ...

**24**

votes

**5**answers

2k 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 ...

**22**

votes

**3**answers

1k 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 ...

**21**

votes

**8**answers

3k 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 ...

**14**

votes

**4**answers

1k views

### Group Structure on CP^infinty

I was inspired by the following algebraic topology orals question:
"Is $S^1$ the loop space of another space?"
This is easy to see if you recognize that $S^1$ is a $K(\mathbb{Z},1)$, and the loop ...

**10**

votes

**3**answers

2k 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. ...

**19**

votes

**1**answer

706 views

### Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups.
Below is a proposition for ...

**16**

votes

**3**answers

2k views

### Which groups have only real and quaternionic irreducible representations?

Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:
1) it's not isomorphic to its dual ...

**10**

votes

**6**answers

2k views

### Polynomial invariants of the exceptional Weyl groups

Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let ...

**9**

votes

**4**answers

2k views

### Exceptional isomorphisms of Lie groups

It is known that in low dimensions certain exceptional isomorphisms arise between Lie groups. I have read about some of them in some papers, but I have not been able to find a "systematic" treatment ...

**13**

votes

**0**answers

336 views

### Nilpotent pro-$p$ groups

Is it true that every finitely generated (topologically) torsion free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...

**10**

votes

**3**answers

3k views

### Curvature of a Lie group

Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold has some scalar curvature R. Is there a nice formula which relates the lie algebra of the ...

**14**

votes

**3**answers

704 views

### There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?

How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) ...

**11**

votes

**3**answers

2k views

### What is the outer automorphism group of SU(n)?

All the automorphisms of $SU(2)$ seem to be inner, which would mean that $\mathrm{Out}$ $SU(2)$ is trivial. Is that correct? Is this true in general $SU(n)$? I can't quite see -- any thoughts would be ...

**9**

votes

**2**answers

904 views

### If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie groups?

Let $G_{1}$ and $G_{2}$ be compact connected Lie groups.
If $G_{1}$ and $G_{2}$ are homeomorphic as topological spaces, are
they isomorphic as Lie groups?

**10**

votes

**4**answers

1k views

### Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$.
An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...

**9**

votes

**3**answers

2k views

### Left invariant metric on SLn

I am looking for a left invariant metric on $SL_n(\mathbb{R})$. If this is not possible, it would be acceptable to have a metric on $SL_n(\mathbb{R})/SO_n(\mathbb{R})$ or something like that. Is there ...

**9**

votes

**2**answers

990 views

### Christoffel symbols on a Lie group in Riemann normal coordinates

Consider a coordinate patch around the identity element in a Lie group given by the exponential mapping (Riemann normal coordinates). We have a Levi-Civita connection corresponding to the ...

**7**

votes

**1**answer

626 views

### Differences in philosophy between Lie Groups and Differential Galois Theory

As far as I have heard,Sophus Lie's aim was to construct an analogue of galois theory for differential galois theory. I am familiar with lie group but not with differential galois theory. What is the ...

**7**

votes

**3**answers

2k views

### Matrix expression for elements of $SO(3)$

Hi all. Is there any explicit matrix expression for a general element of the special orthogonal group $SO(3)$? I have been searching texts and net both, but could not find it. Kindly provide any ...

**8**

votes

**2**answers

746 views

### Cohomology of lattice subgroups

I am trying to find a reference for lower cohomology groups $H^i(G, \mathbb{Z}),$ for $i=1, 2, 3$ for lattices in higher rank (for example, $SL(n, \mathbb{Z}), Sp(2n, \mathbb{Z}),$ and possibly ...

**7**

votes

**0**answers

493 views

### Small sum of group elements acting as rank 1 matrix.

I am interested in constructing small (as possible) group $G$ with large dimensional irreducible representation $\rho,V$ such that exist three elements of $g_1,g_2,g_3\in G$ such that for some ...

**6**

votes

**2**answers

547 views

### Ricci curvature of the symplectic group

Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a ...

**4**

votes

**1**answer

390 views

### Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...

**3**

votes

**1**answer

439 views

### The (-)-Connection on a Lie Group

Is the geodesic exponential map for a Lie group with the (-)-connection a diffeomorphism? This connection is one of two flat connections introduced by Cartan and Schouten on a Lie group and has ...

**15**

votes

**1**answer

366 views

### Weyl group actions on 0-weight spaces

For a complex simple Lie group G with a maximal torus T, we can take a highest-weight representation V of G and look at the 0-weight space, i.e. the subspace of V of elements invariant under T. This ...