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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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Is an action of $\mathbb{S^1}$ on itself effective if and only if it is free?

Obviously if the action is free it is effective. In the case of $\mathbb{S^1}$ acting on itself by rotations/complex multiplication this is clearly the case, but are there other possible actions of $\mathbb{S^1}$ on itself which I haven't…
R Mary
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Are there any good ways to see the universal cover of $GL^{+}(2,\mathbb{R})$?

Are there any good ways to see the universal cover of $GL^{+}(2,\mathbb{R})$? Here $GL^{+}(2,\mathbb{R})$ stands for the identity component of $GL(2,\mathbb{R})$, i.e. positive determinant matrices. I am looking for an explicit description of…
M. K.
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The Lie algebra of a group of matrices

How does one find the Lie Algebra of a Lie group G which is given by matrices $$\left(\begin{array}{cccc} \cos \theta & -\sin \theta & x & y \\ \sin \theta & \cos \theta & z & w \\ 0 & 0 & \cos \theta & -\sin \theta \\ 0 & 0 & \sin \theta & \cos…
Luc
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Showing that the commutator subgroup of a Lie group is a Lie subgroup

I'm learning about Lie groups and Lie algebras independently, and I'm trying to show that the commutator subgroup, $H=[G,G]$, of a Lie group, $G$, is a Lie subgroup. My first instinct was to take a look at the top down approach of defining…
Jimmy B.
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Dimension of SO(n) and its generators

The generators of $SO(n)$ are pure imaginary antisymmetric $n \times n$ matrices. How can this fact be used to show that the dimension of $SO(n)$ is $\frac{n(n-1)}{2}$? I know that an antisymmetric matrix has $\frac{n(n-1)}{2}$ degrees of freedom,…
sagsg dagdgah
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Closedness of connected semisimple Lie subgroups of semisimple groups

A connected semisimple Lie subgroup of $SO(n)$ is closed in $SO(n)$ (Kobayashi and Nomizu, 1963, p. 279). Can we extend this result to all semisimple groups, put differently, is any connected semisimple Lie subgroup of a semisimple group closed in…
user1043065
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Is $U(2)=SU(2) \times U(1)$?

In the many textbook of standard model, i encounter the relation \begin{align} SU(2)_L \times U(1)_L = U(2)_L \end{align} Here $L$ means the left-handness, (It is a physical meaning(representation), which states that fermion have left or right…
phy_math
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Compact Lie group with non discrete center?

Could someone please give me an example of a compact Lie group with non discrete center which is not just a product of a group with a torus?
Huey p Newton
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What's wrong with this trivial proof that every element of a compact Lie group is contained in a maximal torus?

The Lie groups book I'm reading (Knapp, Lie Groups Beyond an Introduction, page 255) goes to some trouble to prove that every element of a compact Lie group is contained in a maximal torus. Why isn't this obvious from the fact that every element is…
dessin d'enfant terrible
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Sets of orthogonal matrices are bounded

I have already shown that $O(n), SO(n), U(n), SU(n)$ and $Sp(n)$ are closed. Now I want to show that they are bounded. But when I tried, I noticed I need a metric or a norm on these sets. But there are several possibibilities to define a norm on…
learner
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Proof help: $SU(2)$ is a double cover of $SO(3)$

I am reading a proof that $SU(2)$ is a double cover of $SO(3)$. My source is this set of notes: http://www.damtp.cam.ac.uk/user/examples/D18S.pdf. The proof begins near the bottom of page 4. I have had no trouble with most of the proof, but I am…
Gyu Eun Lee
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Correspondence between one-parameter subgroup and left-invariant vector field.

Given a one-parameter subgroup of a Lie group $G$, we can show that the one-parameter subgroup $\phi : \mathbb{R} \to G$ defines a unique left-invariant vector field $X: \frac{d\phi^\mu(t)}{dt} = X^\mu(\phi(t))$. However, we know that even in an…
Quantization
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More elegant proof of that this diagram commutes

Define an $\mathbb R$-linear isomorphism $f_n : \mathbb C^n \to \mathbb R^{2n}$ by $(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$. Let $A \in M^n (\mathbb C)$ and define $R_A(v) = v \cdot A$. Furthermore define an injective…
learner
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Textbook literature on Lie groups

I'm a student that wants to get to know Lie groups. I know a bit about manifolds and a bit about groups, but nothing about topological groups or such things. Can you suggest a textbook that covers the matter rather explicitly and in rather basic…
Stefan Hante
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Subgroup of $GL_{n+m}(\mathbb K)$

Let $G$ be a subgroup of $GL_n(\mathbb K)$ and $H$ a subgroup of $GL_m(\mathbb K)$ where $\mathbb K \in \{\mathbb R, \mathbb C, \mathbb H\}$. I want to prove that there exists a subgroup of $GL_{n+m}(\mathbb K)$ isomorphic to $G \times H$. Can you…
learner
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