MathOverflow Stack Exchange
MathOverflow Stack Exchange

Questions tagged [lie-groups]

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

2945 questions
22
votes
1 answer

Can we realize Weyl group as a subgroup?

Given a semisimple Lie group G, let T be a maximal torus, W be the Weyl group defined as the quotient N(T)/C(T), where N(T) denotes the normalizer of T and C(T) denotes the centralizer. Two questions are: How many ways are there we can realize W…
user1832
  • 2,679
15
votes
0 answers

Maximal Tori and group structures on spheres

It is known that for any compact Lie group $G$ with maximal torus $T$, that any other maximal torus $T'$ is conjugate to $T$. This might be a bit of a stretch, but I was wondering if it is possible to use this result to deduce which positive…
Geoffrey
  • 727
  • 6
  • 14
13
votes
3 answers

Lie subgroups of SU(3)

Apart from images of representations of subgroups of SU(2), what are the Lie subgroups of SU(3)? Where should I look for a reference?
Alfred Wood
  • 131
13
votes
3 answers

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 $\mathrm{Hom}(K,G)$ is the set of Lie group homomorphisms,…
Charles Rezk
  • 26,634
12
votes
4 answers

Is an identity that is true for matrix Lie groups true for all Lie groups?

Many identities for Lie groups are more easily proved for matrix groups. A non-trivial example is the equation $$ \frac{d}{dt}\vert_{t=0} \exp(-X)\exp(X+tY) = \frac{1-e^{-\operatorname{ad} X}}{\operatorname{ad} X} Y. $$ My question is if it is…
Eric O. Korman
  • 3,204
12
votes
1 answer

Are infinite dimensional Lie algebras related to unique Lie groups?

For every finite dimensional Lie algebra $g$, there is a unique simply-connected Lie group $G$ whose Lie algebra is $g$. Is this true in the infinite dimensional case?
Ramand
  • 317
  • 1
  • 10
12
votes
2 answers

"No Small Subgroups" Argument

What is the "no small subgroups" argument for $GL(n,\mathbb R)$? That is, how do we show that in $GL(n,\mathbb R)$ there exists a neighborhood of the identity which contains no subgroup other than the trivial one? I had some scribbling (for the…
Murat Güngör
  • 486
11
votes
3 answers

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 references.
mathstudent
  • 121
  • 1
  • 4
11
votes
3 answers

Can we promote to a Lie Group Isomorphism?

We regard an isomorphism of Lie groups to mean a group isomorphism which is simultaneously a diffeomorphism of the underlying smooth manifold. I'm wondering about how much rigidity is imposed by this definition. Question: If we have maps $f, g:…
Jon Cohen
  • 1,251
9
votes
5 answers

Occurrence of semi-spin groups

In the classification of simple Lie algebras one has the familiar picture of 4 families, $A_n$, $B_n$, $C_n$ and $D_n$, and 5 exceptional groups, $F_4,$ $G_2,$ $E_6$, $E_7$ and $E_8$. The $D_n$ family has the unique feature that it contains, among…
Johan
  • 616
  • 3
  • 9
9
votes
1 answer

Cohomology of the unitary group

The de-Rham cohomology ring of U(n) is the exterior algebra generated by the odd-dimensional classes x_1, x_3, ..., x_(2n-1). Moreover, on a Lie group every cohomology class is represented by a unique invariant form (both left and right). I ask two…
Fabio
  • 1,192
9
votes
2 answers

Group G hasn't all conditions of Lie group

Is there a group $G$ with the property that $G$ is a smooth manifold, the multiplication map of $G$ is smooth, but the inversion map of $G$ is not smooth?
R Salimi
  • 221
  • 1
  • 7
8
votes
2 answers

Compact connected Lie groups isomorphic as groups and manifolds

Let $G_1$ and $G_2$ be compact connected (not necessarily semi-simple) Lie groups. Assume that the underlying smooth manifolds of $G_1$ and $G_2$ are diffeomorphic and that the underlying abstract groups are isomorphic. Is it true that $G_1$ and…
rori
  • 231
8
votes
2 answers

Do all closed connected subgroups of $SO(2n+1)$ embed into $SO(2n)$?

Is every closed connected proper subgroup of $SO(2n+1)$ isomorphic (as a Lie group) to a subgroup of $SO(2n)$? The answer is yes for abelian closed connected subgroups.
Igor Belegradek
  • 28,360
7
votes
2 answers

Question on KAK decomposition

Let $G$ be a semisimple Lie group and let $G = KAK$ be a Cartan decomposition. For $\mathrm{SL}_2(\mathbb{R})$ it holds for every $g \in G$ that $KgK = Kg^{-1}K$. Does the same hold for every semisimple Lie group?
Constantin K
  • 419
1
2 3 4 5 6