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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 $\frac{n(n-1)}{2}.$

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Is $\mathfrak{s}\mathfrak{u}(d) \otimes \mathfrak{s}\mathfrak{u}(d)=\mathfrak{s}\mathfrak{u}(d^2)$?

Given a basis $a_i$ for the lie algebra $\mathfrak{s}\mathfrak{u}(d)$, does the set of elements $a_i \otimes a_j$ form a basis for $\mathfrak{s}\mathfrak{u}(d^2)$?
Joel Klassen
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What is $D_{5}$

I have recently encountered a Lie group in a paper called $D_{5}$ as a subgroup of $E_{6}$. I have tried googling with mixed results. Is it just $\operatorname{SO}_{10}(\mathbb C)$?. Is it $\operatorname{Spin}(10)$? I am looking for geometric…
ShotaG
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Diffeomorphism in Lie Group

$G$ is a Lie group and consider $L_{g}: G \rightarrow G$ ($L_g(h)=gh$). What i need to show that $L_{g}$ is diffeomorphism. Is it something obvious? Can someone explain it to me?
cactus
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Does the projection from a compact Lie group to its component group split?

This is an elementary question that probably admits an elementary counterexample, but ... Let $G$ be a compact Lie group and $G_0$ its identity component. One then has a short exact sequence $$ 1 \to G_0 \to G \to \pi_0(G) \to 1; $$ the last object…
jdc
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Intuitive understanding of lie group definition

So I have the following definition from the book: Definition: A matrix Lie group is any subgroup $G$ of $GL(n, \mathbb{C})$ with the following property: If $A_m$ is any sequence of matrices in $G$, and $A_m$ converges to some matrix $A$ then either…
user111750
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Lattices in $PSL_2(\mathbb{R})$

Can a lattice in $PSL_2(\mathbb{R})$ have a normal abelian subgroup? It looks to me that it doesn't, but where can I read a proof?
student
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How can I show that this matrix is a flip

I am trying to show that $$ F = \left(\begin{array}{cc} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array} \right ) $$ is a flip about a line through the origin. What I tried: Let $v \in \mathbb R^2$, $v = (r \cos \varphi, r \sin…
learner
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What Lie groups have a discrete set of order two elements?

We know that the set of order two elements of $R^n$, tori and $S^3$ are discrete. Are there others examples of Lie groups with such property? Are there some characterization of such class?
Daniel Vendruscolo
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Question on $\mathrm{Ad}(SL(3,\mathbb R))$

The following question appears in an example in page 116, Representation theory of semisimple groups, A. W. Knapp. Consider $G=\mathrm{Ad}(SL(3,\mathbb R))$. It is a subgroup of $GL(8,\mathbb C)$. Let $U$ be the compact real form of $G$. Then $U$…
Qinghua Pi
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Do the infinitesimal generators of a group need be necessarily exponential

A vector field is supposed to be an infinitesimal generator of a lie subgroup. Typically to generate a flow on a manifold using a one parameter group of transformations, we do exponentiation of the vector field. Does it have to be necessarily…
pencil
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$A_{2}^{T} + A_{2} < 0$ for $A_{2} =(A_{1}A_{0}^{-1})^{\alpha}A_{0}$?

Given are two matrices $A_0, A_1$, whose symmetric part is negative definite: $A_{0}^{T} + A_{0} < 0$, $A_{1}^{T} + A_{1} < 0$ How could one proof that: $A_{2}^{T} + A_{2} < 0$ for $A_{2} = (A_{1}A_{0}^{-1})^{\alpha}A_{0}$ and $\alpha \in [0,…
Lyapunov123
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$Sl(2,\mathbb{R})$ and matrix exponential

I'm trying to prove that every matrix in $Sl(2,\mathbb{R})$ can be written as a product of two exponential matrix. First I noted that every matrix in $Sl(2,\mathbb{R})$ can be written as a product of a orthogonal matrix and a upper triangular…
User43029
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How to visualize 2D or 3D Rotation matrices

How to visualize 2D or 3D Rotation matrices in $\mathbb{R}^3$ or $\mathbb{R}^2$? If so, can we preserve the manifold properties? Like geodesic distance?
gsoldier
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A definition of SU(2) using only Lie parameters and the product/inverse functions (i.e group proprieties ), not 2x2 complex matrices

If I define a Lie groupe as an analytical manifold of dimension N with the following group operations: $g(a) g(b) = g(p(a,b))$, $g^-1(a) = g(r(a))$ Wenn labelling elements of the group $a=(a_1, …, a_N), b=(b_1, …, b_N)$, with p and r analytic, I am…
Matthieu Giriens
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Affine transformation Group $\operatorname{Aff}(1)$ is the only connected, non-abelian, 2D Lie group

I'm reading Stillwell's Naive Lie Theory. On page 88 he says, (The affine group of the line) $\operatorname{Aff}(1)$ is in fact the only connected, nonabelian two-dimensional Lie group. May I ask how to prove this?
athos
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