Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [lie-groups]

A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group.

Consider using with the (group-theory) tag.

Lie groups are groups that are also differentiable manifolds that represent the best developed theory of continuous symmetry of mathematical objects.

Examples of lie groups are:

1) The Euclidean space $\mathbb{R}^n$ under addition is a lie group.

2) The special orthogonal group of real orthogonal matrices with determinant $1$ (note that $n=3$ is the rotation group in $\mathbb{R}^3$).

3) The spin group, which is the double cover of the special orthogonal group such that $\exists$ a sequence of lie groups:

\begin{equation*} 1\to Z_2\to~\text{Spin}(n)\to SO(n)\to 1. \end{equation*}

Note that it has dimension $\frac{n(n-1)}{2}.$

7686 questions
5
votes
2 answers

Is there a relation between the connected components of a Lie group?

Let $G_0$ be the identity component of a (compact) Lie group $G$ and $G_1$ one of its connected components. Is there any relation between $G_0$ and $G_1$? For example, is there a $g\in{G_1}$ such that $L_g\left(G_0\right)=G_1$? Here $L_g$ is the…
user3257624
  • 361
5
votes
1 answer

confused in the term "closed" in closed subgroup

well, In Brian C Halls Book, I am not getting the definition of Matrix Lie group, as he says : A matrix Lie Group is any subgroup $G$ of $GL_n(\mathbb{C})$ with the following property: If $A_m$ is any sequence of matrix in $G$ and and $A_m$…
Myshkin
  • 35,974
  • 27
  • 154
  • 332
5
votes
1 answer

How to write down the pull back of a differential form by exponential map?

The exponential map $e_{m}: M(n,\mathbb{R})\rightarrow M(n,\mathbb{R})$ is defined by $$e_m(\alpha)=me^\alpha,\quad e^\alpha=1+\alpha+\frac{\alpha^2}{2}+\frac{\alpha^3}{3!}+\cdots$$ Now fix $q\in M(n,\mathbb{R})$ and introduce the left-invariant…
Bombyx mori
  • 19,638
  • 6
  • 52
  • 112
5
votes
1 answer

Is a finite volume Lie group compact?

I know an example of a finite volume homogeneous space which is not compact, $SL_2(\mathbb(R)) / SL_2(\mathbb{Z})$. But what about a Lie group with this property? Can it happen? (The Lie group is assumed to have the Haar measure.)
Elle Najt
  • 20,740
5
votes
2 answers

Does every non-abelian Lie group have a finite subgroup?

I am interested in finite-order elements (different from the identity) of non-abelian Lie group. It seems to me that each non-abelian Lie group has at least one (actually many) finite-order elements or, in other terms, one or more finite subgroups.…
Nick Belane
  • 105
5
votes
1 answer

why is $S^2$ not a Lie group?

I'm reading John Stillwell's "Naive Lie Theory" and it was mentioned there (without giving a proper definition of what a Lie group is) that the only Lie groups among the unit n-spheres are $S^1$ and $S^3$. Is there a simple or naive explanation of…
Nathan Sikora
  • 1,165
  • 8
  • 18
5
votes
1 answer

Generators of the SU(2) matrix group

Let $X_i$ be the generators of $SU(2)$ and let the parameters of the rotation be $\theta, \phi, \delta$ such that the matrix $R = e^{i(\phi X_{1} + \delta X_{2} + \theta X_{3})}$, where $R$ is an element of $SU(2)$. So, $R = 1 + i(\phi X_{1} +…
nightmarish
  • 617
5
votes
0 answers

What is the center of SO(p,q)?

For the matrix lie group $SO(p,q)$, what is in the center? Is there anything other than $I_n$ (or $-I_n$ in the case $p+q=n$ is even)? Also where can I find a reference?
Michelle
  • 185
5
votes
1 answer

Typo on Wikipedia? Dimension of $U(n)$

Let $U(n)$ denote the unitary group. That is, $$ U(n) = \{A \in GL_n(\mathbb C)\mid A^\ast A = I\}$$ Wikipedia states: "The unitary group $U(n)$ is a real Lie group of dimension $n^2$. " There seem to be two typos: one, unitary matrices are complex.…
learner
  • 1,967
5
votes
0 answers

Proof that $GL_n, SL_n$ are not bounded

Please could someone check my work on this exercise (from book I am reading). Thanks! Exercise: Prove that $GL_n (\mathbb K)$ is non-compact when $n \ge 1$. Prove that $SL_n (\mathbb K)$ is non-compact when $n \ge 2$. What about $SL_1 (\mathbb…
learner
  • 1,967
4
votes
2 answers

How to prove that the adjoint group is a Lie subgroup of $Gl(\mathfrak{g})$

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $Ad: G \rightarrow GL(\mathfrak{g})$ be the Adjoint representation. I want to prove that $Ad(G)$ is a Lie subgroup of $GL(\mathfrak{g})$. Here is what I tried: From Cartan's Theorem, it…
Bruno Suzuki
  • 481
4
votes
2 answers

complexification of $SO(2)$

While computing the complexification of Lie group $SO(2)$, I get the result is all the matrix of the following form $$\left(\begin{array}{cc} \frac{e^{t-\sqrt{-1}\theta}+e^{-t+\sqrt{-1}\theta}}{2} &…
Daniel
  • 319
4
votes
1 answer

$[X,Y]=0 \implies \exp(X+Y)=\exp(X)\exp(Y)$

I am trying to show that if $[X,Y]=0$ then the exponential map $\exp : Lie(G)\to G$ is such that $$\exp(X+Y)=\exp(X)\exp(Y), \forall X,Y\in Lie(G).$$ The hint is to show that $\gamma : t\mapsto \exp(tX)\exp(tY)$ is a one-parameter subgroup, which I…
roi_saumon
  • 4,196
4
votes
1 answer

How to prove that every Lie group is the semidirect product of a connected Lie group and a discrete group?

Every Lie group is the semidirect product of a connected Lie group and a discrete group. I think the component of the identity could be useful.
Gsanm
  • 49
4
votes
2 answers

The map from $SU(2) \times SU(2)$ to $SO(4)$

Lie group $SO(4)$ is doubly covered by $SU(2) \times SU(2)$, I want to know the map from $SU(2) \times SU(2)$ to $SO(4)$. The map from $SU_{2}$ to $SO(3)$ is $\begin{pmatrix} \alpha & \beta \\-\overline{\beta} & \overline{ \alpha} …
guojm
  • 345
1 2 3
…
19 20