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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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Classify the compact abelian Lie groups

It's a classical theorem of Lie group theory that any compact connected abelian Lie group must be a torus. So it's natural to ask what if we delete the connectedness, i.e. the problem of classification of the compact abelian Lie groups.
Lao-tzu
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Non-algebraic Lie groups

When trying to learn about Lie groups I find that most natural examples of Lie groups are actually examples of algebraic groups. What are some interesting examples of Lie groups which are not algebraic groups?
the L
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Elements of order 2 in a Weyl group.

I would like to prove that any element of order 2 in a Weyl group is the product of commuting root reflections. I am told that the proof should be by induction on the dimension of the -1 eigenspace. Any clue would help, thanks !
Vincent
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Left-invariance of differential forms vs left-invariance of vector fields

Given a Lie group $G$, we say that a vector field $X$ is left-invariant if for each $g\in G$ we have $\mathrm{d}L_g(X)=X$, that is $\mathrm{d}L_g|_h(X_h)=X_{gh}$ for all $h\in G$, where $L_g\colon G\to G$ is the left translation by $g$. We also have…
yellon
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Prove that $O(n)$ is a maximal compact subgroup of $GL(n,\mathbb R)$

The indication is: Let $P$ is a symmetric positive definite matrix such that the norm of $p^k$ is smaller than a constant $C$ for every integer $k$, then $P=I_n$.
PAM
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On the Volume of Compact matrix Lie groups

When we define the volume of a compact matrix lie group (subgroup of $M_n(C)$) by viewing it as a subspace of $R^{n^{2}}$ and applying the usual Lebesgue measure, what's the volume of SO(n), SU(n), Sp(n) and...?
Jun Su
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Are there nonsmooth Lie groups?

The definition of "Lie group" typically restricts to a smooth manifold. If we instead define a "Lie group" to be a topological manifold such that multiplication and inversion are continuous, is the manifold necessarily smooth? Is the smooth…
Geoffrey Irving
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Complexification and universal complexification of a Lie group

Not all real Lie groups have a complexification, but the universal complexification always exists and is unique. My question is, when is a complexification also the universal complexification? Edit: Let me make my question more precise: Let $G$ be a…
Qidi
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Rotation matrices and $SO(3)$

Let $R$ denote the set of matrices that rotate $\mathbb R^3$ around an axis. For the $x,y,z$ axes the matrices are given here. Let $SO(3)$ denote the set of orthogonal $3\times 3$ matrices with determinant $1$. It is clear to me that $R \subseteq…
self-learner
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On compact, simply connected Lie group and its subgroup

Let $G$ be a compact, simply connected Lie group, and $H$ is its Lie subgroup that is also compact and simply connected, and has the same dimension with $G$, then should $H=G$? Note that these imply that the Lie group $G$ and $H$ are locally…
Lao-tzu
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Does the algebraic structure of a Lie group restrict the possible dimensions of other Lie groups isomorphic to it?

In a recent question, I initially doubted that $\mathbb{C}^\times\cong S^1$, my intuition being that $\mathbb{C}^\times$ has one more "dimension" than $S^1$ - in rigorous terms, $S^1$ is (or rather, can be given the structure of) a 1-dimensional Lie…
Zev Chonoles
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Lie groups locally isomorphic to $\operatorname{SL}_2(\mathbb{R})$

I was reading "Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces." They prove a certain theorem for Lie groups locally isomorphic to $\operatorname{SL}_2(\mathbb{R})$ which are connected and have finite center. But they…
user147556
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Exponential map to simply connected abelian Lie group is an isomorphism.

In my notes I have the following claim. Let $G$ be a simply connected abelian Lie group. Since $G$ is abelian, we have that $exp(A+B) = exp(A) exp(B)$ for all $A,B∈Lie(G)$. Therefore, the exponential map is an isomorphism between…
roi_saumon
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Show that every compact Lie group contains a finitely generated dense subgroup.

I'm trying to show that the connected component has a finite number of distinct maximal tori. So the group generated by its generators must be dense. But I don't know if it is really true.
Andre Gomes
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Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}=H$ for compact Lie groups?

Let $H$ be a closed subgroup of a compact Lie group $G$. Suppose that $gHg^{-1}\subseteq H$ for some $g\in G$. Does it follow that $gHg^{-1}=H$? If $H$ is connected the answer is clearly yes since $gHg^{-1}\subseteq H$ implies…
Simon Parker
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