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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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Lie Groups of bigger cardinality

A Lie Group is to be a group that is also a manifold, and of course a manifold is a second countable Hausdorff space. Now the maximum cardinality for a second countable (Hausdorff)space is $\beth_{2}$ = $\mathcal P^{2} \left({\mathbb{N}}\right)$.…
Mr X
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Decreasing sequence in subgroup

Can someone clarify the following statement: Let $H\subset\mathbb{R}$ be a subgroup $\neq 0$. Let $a = \text{inf} \lbrace x\in H \vert x>0 \rbrace$. If $a\notin H$ then there exists a decreasing sequence in $H$ converging to $a$. What guarantees…
guestguest
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The trivial subgroup is closed (in the sense of topology)

I try to understand the proof of the below statement: Statement: Let $\phi: G \rightarrow H$ be a homomorphism of Lie groups. Then the kernel of $\phi$ is a closed subgroup of $G$. Proof: Put $K= $ Ker $\phi$. Then $K$ is a subgroup of $G$. Now…
guestguest
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Is my proof that $\operatorname{Isom_n}{\mathbb R^n}$ is compact right?

I previously did this exercise: Prove that $\operatorname{Isom_n}{\mathbb R^n}$ is a matrix group. where $$\operatorname{Isom_n}{\mathbb R^n} = \left \{ \left ( \begin{array}{cc} A & 0 \\ V & 1 \end{array} \right ) \mid A \in O(n) (\mathbb K), V…
learner
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These subsets of $O(n)$ are clopen

Please could someone check my work on this exercise (from a book I am reading). Thanks! Exercise: Prove that $SO(n)$ and $ O(n)^- = \{ A \in O(n) \mid \det(A) = -1 \}$ are both clopen in $O(n)$. My solution: Since $\det: O(n) \to \{-1,1\}$ is…
learner
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Isn't this $f$ always a group isomorphism

Consider the following exercise from a book I'm reading: If $n$ is odd show that $$ f: O(n) \to SO(n) \times \{1,-1\}, A \mapsto (A \operatorname{det}{A}, \operatorname{det}{A})$$ is an isomomorphism. But why does $n$ have to be odd? I can easily…
learner
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Topology of orthogonal groups when $n > 4$?

On Wikipedia I read that the topologies of $O(1)$ and $SO(1)$ to $SO(4)$ are known topological spaces. What about $O(n), SO(n), U(n)$ when $n>4$?
learner
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Elements in finite symmetry groups share a common fixed point

I am reading Tapp's intro to matrix groups for undergraduates. On page 46 he states the following theorem: For $X\subseteq \mathbb R^2$ if $Symm(X)$ is finite then it is isomorphic to $D_m$ or $\mathbb Z_m$ for some $m$. Following it he writes: The…
learner
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Norm preserving matrices also preserve inner products

I am trying to prove that if $A \in M_n(\mathbb C)$ preserves norms then it also preserves inner products. I showed this for real matrices and I want to use this for this proof here. Let $f_n: \mathbb C^n \to \mathbb R^{2n}$ be the map $(x_1 + iy_2,…
learner
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On the definition of quaternionic-linear real matrices

I'm reading Tapp's introduction to matrix groups. The book introduced complex-linear matrices. Let me reproduce the definition in my own words: Let $B\in M_{2n}(\mathbb R)$. Let $J$ be the matrix $$ J = \left (\begin{array}{} 0 & -I \\ I &…
learner
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More on rotation matrices: a basis for $SO(3)$?

Consider the group $SO(3)$. The rotation around the $x$-axis is represented by the matrix $$R_x = \left ( \begin{array}{ ccc } 1 & 0 & 0 \\ 0 & \cos \Theta & - \sin \Theta \\ 0 & \sin \Theta & \cos \Theta \end{array}\right ) $$ Similarly, for the…
self-learner
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Matrix form of $SU(2)\times SU(2)$ acting on the Quaternions

I have been given a function $\varphi(A,B):\mathbb{H}\to\mathbb{H}$, $h\mapsto AhB^{-1}$ where $A,B\in SU(2)\times SU(2)$. I don't understand how this forms a well-defined map, nor how this would even return quaternion. Going off the answer given…
Thomas Pluck
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Dimension of Lorentz group

Consider the Lorentz group $SO(1, n)$. I would be interested in knowing the dimension of this Lie group for general $n$. Can you tell me, what its dimension is? Thank you in advance!
Niklas
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What background is needed to self-study/learn lie group? Which do books recommend?

What background is needed to self-study/learn lie group? Which do books recommend?
cayic
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Subgroups of the group $G_2 \times G_2$

Does the group $G_2 \times G_2$ have the group $SO(7)$ (or its double cover $Spin(7)$) as its subgroup? Here, $G_2$ is the compact exceptional group $G_2$.
Arash Ranjbar
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