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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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Nightmares about computers and Lie algebras

I'm writing some code that does simulations of quantum field theory, and I came to the sudden realization that I hadn't yet written the part for the SU(n) Lie group. I know this group is a subgroup of U(n) and GL(n), but GL is already infinite and U…
richyfeynman
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How to find coordinates on the quotient

I need to work on the quotient space $SL(2,\mathbb R)/SO(2,\mathbb R)$ and I am having trouble finding coordinates on this space. Any help would be really appreciated.
Yang S.
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The special unitary group is a Lie Group

I am trying to find a proof of the fact that the special unitary group is a Lie Group, but I can't come up with any good ideas and I couldn't find anything by searching at google. Could you please help me? Thank you in advance
perlman
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Why the closure is also abelian?

Let be $G$ a Lie groupe and $H$ a Lie subgroup,if $H$ is abelian then the closure $\bar{H}$ of $H$ is also abelian. Please give a proof of this proposition.
Antema Kely
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Subgroup generated by an element of infinite order in a compact group

Suppose one has a connected compact Lie group $K$, and an element $k\in K$ of infinite order. Then can the subgroup generated by $k$ be discrete? I suspect the answer is no, but this suspicion is mostly based on the example of irrational rotations…
geometricK
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$G$ is a compact nontrivial Lie group with a bi-invariant metric, is $\{g \in T_eG|4 \geqslant |g|>1\}$ a lie subalgebra?

$G$ is a compact nontrivial Lie group with a bi-invariant metric, is $\{g \in T_eG|4 \geqslant |g|>1\}$ a lie subalgebra? I doubt that this would be true since it might not be closed under the lie bracket.
Keith
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Compact semismple Lie group.

Let $G$ be a connected Lie group with a compact semisimple Lie algebra. It is well known that the Lie algebra then can be written as the product of compact simple Lie algebras which are classified by Dynkin diagrams. I would like to know what we can…
Niklas
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Discrete centre of Lie group

Can the centre of a (compact, semisimple) Lie group $G$ be discrete? If yes, under which conditions on $G$ this happens? Any references would be appreciated.
user3257624
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One parameter subgroups of a Lie subgroup

Suppose $H$ is a closed connected Lie subgroup of $G$. Are one parameter subgroups of $H$ also one parameter subgroups of $G$? If not true in general, when is the statement true?
Mohammadreza Bidar
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How to get homomorphism from $\mathfrak{U}(\mathfrak{g}) \to Diff(M)$ from an action $G \curvearrowright M$

Given a lie group $G$ and an action $G \curvearrowright M$ where $M$ is a manifold, how can I get a homomorphism $\mathfrak g \to Diff(M)$. Here $Diff(M)$ is the universal differential operators on $M$. My professor wrote down this step on the…
De Yang
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Number of generators of O(N)?

If $SO(N) $ is the connected subgroup of $O(N)$ that contains the identity, is it meaningful to discuss generators of $O(N)$? Can we represent elements of $O(N)$ as the exponential of some quantity in the Lie Algebra of $SO(N)$ or otherwise? Or does…
Eweler
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Lie algebra of $\operatorname{GL}_n\mathbb{R}$

Let $G=\operatorname{GL}_n\mathbb{R}$. I am trying to understand why $L_G=M_n\mathbb{R}$ with the usual Lie algebra structure of $M_n\mathbb{R}$. I understand the canonical identification $L_G=T_1G=M_n\mathbb{R}$. I understand that a tangent…
Terry
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Calculation of commutator of Lie algebra for affine linear maps

This problem was answered before, but I'm stack with a technical point. Let $G$ be the Lie group of linear polynomials under composition (that is, affine transformations), $$\{x \mapsto ax+b, a\neq 0, a,b\in\mathbb{R}\}.$$ I'm going to use a dirty…
Math.StackExchange
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Left invariant forms on lie groups?

Let $G$ be a Lie group. A vector field $X\in\mathfrak{X}(G)$ is left-invariant if the diagram below is commutative: for every $g\in G$ where $L_g$ stands for the left translation by $g$. Now a differential $1$-form $\omega\in \Omega^1(G)$ is left…
PtF
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Dimensions of classical Lie groups

I understand that the dimension of the $SU(n)$ matrix group is $n^2$ because there are $2n^2$ real variables (for the $n^2$ complex matrix elements) in each matrix, and there are $n^2$ equations (arising from the unitary condition) that relate the…
nightmarish
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