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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
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How a matrix in SL(2,R) can be seen as a product of symmetric matrix and a rotation one

I'm reading Gilmore's Lie Group, Physics and Geometry and I'm trying to solve the exercise 1 at page 8, in which you have to find the full parametrization of a 2X2 $SL(2,R)$ matrix as a product of a symmetric matrix and a rotation one. While…
riccardoventrella
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Extensions of complex tori by copies of $\mathbb{C}^{\times}$

Let $$1 \rightarrow (\mathbb{C}^{\times})^r \rightarrow G \rightarrow V/L \rightarrow 0$$ be an extension of complex lie groups, where $L$ is a lattice in a complex vector space $V$. Is $G$ necessarily commutative?
safety stegosaurus
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Action of a quotient of a compact Lie Group on M

Let $M$ be a smooth manifold and $G$ a compact Lie Group acting transitively on $M$. Let $H$ be the subgroup of $G$ defined by $\{ g \in G | g \cdot m = m, \forall m \in M \} = \operatorname{Stab}_g(M)$, i.e $H$ is the "global-stabilizer" subgroup…
user7090
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How can a Lie subgroup of a matrix Lie group not be a matrix Lie group

Let $G$ be a matrix Lie group and let $H$ be a Lie subgroup of $G$. My understanding is that all $g \in G$ are matrices, since $G$ is a subgroup of the general linear group. Then aren’t all elements $h \in H$ matrices, since $h \in G$? From what I…
user110971
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Lie group homomorphism from $S^1$ to $S^1$

What are the all the Lie group homomorphisms from $S^1$ to $S^1$? I know that for each $n \in \mathbb{N}$, $z \mapsto z^n$ gives a Lie group homomorphism of $S^1$. Thanks in advance!
Shubham Namdeo
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Two questions on connected nilpotent Lie groups

Let $G$ be a connected nilpotent Lie group. Firstly, I would like to know whether it is true that $G$ is simply connected if and only if its center is. I do not understand how the center of $G$ being simply-connected can imply that $G$ is. Secondly,…
TheoPatr
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Defining Lie groups without the notion of a manifold

I like to introduce (matrix) Lie groups without the notion of manifolds. (And I am happy to sacrify the "few" Lie groups which are not matrix groups in favor of a simpler definition.) I was thinking of the following definition: $G$ is a (matrix)…
B0rk4
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Maximal tori and compact subgroups of $SL_n(\mathbb{C})$?

Let $D\subseteq GL_n(\mathbb{C})$ be the subgroup of diagonal matrices and let $T=D\cap U(n)$. Let us assume that $$ T\subseteq U(n) \subseteq GL_n(\mathbb{C})$$ is a maximal torus and a maximal compact subgroup for $GL_n(\mathbb{C})$. Question:…
Drew Armstrong
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Global fixed point for a compact group action?

Let $G$ be a connected Lie group and let $K\subseteq G$ be a maximal compact subgroup. If $H\subseteq G$ is any compact subgroup I want to show that there exists $g\in G$ such that $g^{-1}Hg\subseteq K$. Proof Idea: Note that $H$ acts on the coset…
Drew Armstrong
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How to find the induced Lie algebra homomorphism

Consider the quaternions $H=\{1+bi+cj+dk, a,b,c,d \in \mathbb{R}\}$ and the norm $\|h\|=\sqrt{h^*h}$, which is a Lie group homomorphism between $H^*$ and $\mathbb R^*$. How can I find the Lie algebra homomorphism induced by the norm?
user43014
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Elements of finite order in compact abelian Lie Group

If $G$ is a compact abelian Lie group, why does the $n$th power map from $G$ to $G$ form a finite covering? I cannot see why the kernel must be finite.
Jesse
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diffeomorphism group of a lie group

Let $G$ be a Lie group, there is a natural inclusion of $G$ in Diff($G$), by left traslation. Is true that there is a splitting Diff($G$)$\cong$ $G$ $\times$ Diff($G,e$)? Where Diff($G,e$) is the subgroup of Diff($G$) that fixes the identity…
Andrés
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Representation of $\mathbb{H}^n$ as a quotient of Lie groups

I would like to know, how to represent $\mathbb{H}^n$ as a quotient $G/H$ of Lie groups (since $\mathbb{H}^n$ is a homogeneous space, such a representation must exist). I have heard that it is possible to represent it as a quotient $SO(1, n)/SO(n)$.…
Nick
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Definition of rank for compact semisimple Lie group

Let $G$ be a compact semisimple Lie group. I have found to different definitions of its rank: One of them defined the rank of the Lie group to be the dimension of a maximal torus. The other definition defined the rank to be the dimension of a Cartan…
Niklas
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Dimensions of submanifolds of SO(n)

I would like to calculate the dimension of \begin{align*} \mathcal{M}_k=\{R\in\mathsf{SO}(n,\mathbb{R})\,|\,\sigma(R)=\{-1,1\},\,m(-1)=k\}, \end{align*} where $\sigma$ is the spectrum and $m$ is the algebraic multiplicity for all…
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