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.
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Is $SU(2) \times SU(2) \times SU(2)$ isomorphic to $U(3)$?
Since $SU(2)$ has dimension 3 and $SU(2) \times SU(2) \times SU(2)$ should have dimension 9 like $U(3)$, right? Or am I missing something?
2
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Spinorial representation of Weyl group of SO(8)
First of all, physicist here, doing my best to post this question as rigorously as possible.
My question is: how can I construct a spinorial representation of the Weyl group of $SO(8)$?
Attempt at a ...
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Cohomology Theory for Flag Varieties
I'm reading these notes about central results of cohomology theory of flag varieties, ie those having the form $G/B$ for $G$ semisimple, simply-connected, complex algebraic group, and $B$ it's (up to ...
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Bounds on the norm of Lie Bracket with respect to Killing Form
If we have a Lie algebra $(V, [,])$ then the Lie bracket is a bilinear map
$$
[, ]: V \times V \to V
$$
Assuming $V$ is finite dimensional, this map is bounded and so, for any choice of norm, there ...
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How to prove Vol_O(n)=2^nVol_O_+(n)?
Let $O(n)$ be the orthogonal group and $DO(n)$ be its discrete subgroup of diagonal matrices with $\pm 1$s on the diagonal.
The metric on $O(n)$ is induced by Euclidian metric on $M(n)$.
Let $O_+(n)=O(...
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Is the kernel of the adjoint representation of a semisimple Lie group always discrete?
Let $G$ be a finite-dimensional, not necessarily compact, semisimple Lie group. Is the kernel $K$ of the adjoint representation $G\mapsto\mathrm{Ad}(G)$ always a discrete subgroup of $G$?
I know that ...
2
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1
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Let $G$ a lie group and $\varphi:G\times G\to G$ given by $(g,h)\to ghg^{-1}$. what is $(d\varphi)_{(g,h)}(X_g,X_h)$.
Let $G$ a lie group and $\varphi:G\times G\to G$ given by $(g,h)\to ghg^{-1}$.
I want to know the derivative of $\varphi$ in the simplest way. I mean, I want to know what is $(d\varphi)_{(g,h)}(X_g,...
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Representation Lie group Heisenberg. [closed]
Let $\mathfrak{H}$ the set of all matrix $m(t,q,p)=\begin{bmatrix} 0 & 0 & 0\\ q & 0 &0 \\ t & p & 0\end{bmatrix}$ (Heisenberg Lie algebra). If $m(t,q,p)\mapsto Rm(t,q,p):=tI+...
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Derivative of an operator exponential.
The image is from the book Lie group by Bryan Hall.
Question. There is some book where 5.10 is valid for operators? for example, I need 5.10 for $X=x$ (multiplication operator by $x$) and $Y=\...
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For a proper Lie group $G$ acting on a complete Riemannian manifold $M$ by isometry, can we find $p'$ close to $p\in M, d(p,p')>d_{M/G}([p], [p'])?$
Let $(M,g)$ be a complete Riemannian manifold, and the Lie group $G$ with $1 \le dim(G) < dim(M)$ acts on $M$ isometrically and properly. Using the metric on $M,$ we define the metric on the ...
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Coframe on coset space via connection one form
Suppose $G$ is a Lie group and $H$ is a Lie subgroup. Then, consider the principal $H$-bundle, $\pi:G\longrightarrow G/H$ where $G/H$ is coset manifold and $\pi$ is the canonical projection. Then, we ...
3
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Concatenation of $f$-related vector fields/tangent vectors
This is probably a highly trivial question, but I just can't wrap my head around it. Let $M,N$ be two manifolds and $f: M \rightarrow N$ smooth. Let further $X^1, X^2$ be two smooth vector fields on $...
1
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1
answer
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Some confusion in Kirillov-Kostant structure
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}=T_e G$. The Kirillov-Kostantstructure Poisson structure on $\mathfrak{g}^*$ is defined as $\{ f,g\} (p)=p([f_{*p},g_{*p}])$ where $f,g\in C^{\...
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Real representations of the symplectic group [closed]
Is there any info on representations of the symplectic groups $ \operatorname{Sp}(n, \mathbb{R}) $ for arbitrary $ n $ ? If not, how would one face the problem of finding representations for these ...
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How to use the contact symmetry group to reduse an order of the ODE? [closed]
When we use a point symmetry with a generator $\hat{X}$, we have to take a group's invariants as a new variables, and we should take some fuction $v$, for which:
$$\hat{X} v = 1,$$
for the last ...
0
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1
answer
43
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The Iwasawa decomposition of $GL(2,\mathbb C)$
On the post (The Iwasawa decomposition of $\text{GL}(2,\mathbf R)$) people said that $\operatorname{GL}(2,\mathbb R)$ has a Iwasawa decomposition of the following form
$$\begin{bmatrix}
\cos\...
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25
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Does every smooth homogeneous manifold have a Lie group acting simply transitively?
I have some smooth manifold $\mathcal{M}$. If there exists a Lie group $G$ that acts transitively on $\mathcal{M}$ does this imply that there exists a Lie group $H$ that acts simply transitively on $\...
2
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Some confusion about $f_{*p}$ when $f\in C^{\infty}(\mathfrak{g}^*)$ and $p\in \mathfrak{g}^*$
Let $G$ be a Lie group with its Lie algebra $\mathfrak{g}$. Let $f\in C^{\infty}(\mathfrak{g}^*)$ and $p\in \mathfrak{g}^*$. We know that $(\mathfrak{g}^*)^*\cong \mathfrak{g}$ and for every $p\in \...
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Why $\frac{d}{dt}(Ad_{e^{tX}}(Y))|_{t=0}=[X,Y]$? [closed]
Let $G$ be a Lie group with its Lie algebra $\mathfrak{g}=T_eG$ and $X\in \mathfrak{g}$. Assume also that $\exp :\mathfrak{g}\to G$ is the eponential map of $G$ which is defined by $X\mapsto \gamma_X (...
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Why $\mathfrak{g}$ is a manifold and $T_p \mathfrak{g}^*\cong \mathfrak{g}^*$? [closed]
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}=T_e G$. Why $\mathfrak{g}$ is a manifold and $T_p\mathfrak{g}^*\cong \mathfrak{g}^*$? Is there a direct isomorphism?
I would really appreciate if ...
0
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0
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10
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Decomposition of a general (not necessarily solvable) real algebraic group $G=(S\times T)\ltimes U$
Let $G$ be a connected real algebraic group. I have seen the following two facts:
Based on this notes
1.If $G$ is nilpotent, then $G_s$ is a closed, connected subgroup (hence a torus). $G_u$ is the ...
0
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0
answers
48
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Why $C(g)$ is a representation of $G$?
I'm trying to study on some lecture notes about $G-$structures of Crainic, but I don't understand the following remark:
We will encounter several quite complicated
vector bundles associated to a $G-$...
4
votes
1
answer
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What geometry is preserved by the translation maps on elliptic curves?
Let $E$ be an elliptic curve (over some field). For any $P \in E$, there is a translation map $T_P: E \to E$ given by $Q \mapsto P+Q$. This map is rational (i.e. the coordinates of $T_P(Q)$ are ...
0
votes
1
answer
37
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Relation between rotation groups and spin groups
Sorry if very dumb question. What exactly is the relation between rotation groups and spin groups? I've heard that the spin groups are the double cover of rotation groups but ks there a more precise ...
1
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1
answer
73
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Why $\phi_{*0}(\frac{d}{dt}|_0)=\frac{d\phi^{\mu}(t)}{dt}|_0\frac{\partial}{\partial g^{\mu}}|_e$?
Let $G$ be a Lie group and $\phi :\mathbb{R}\to G$ be a smooth homommorphism. I know that $\phi_{*0}:T_0 \mathbb{R}\to T_eG$ is a linear map, $T_0 \mathbb{R}=\langle \frac{d}{dt}|_0\rangle $, and $T_e ...
1
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1
answer
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Why specifying values on a local section is enough to determine the local values of a tensorial form on a principal $G$-bundle?
Let $\pi: E \rightarrow B$ be a principal $G$-bundle.
I have an initial context from which my question comes and I will explain how; I believe what I need is more general, but I may be wrong, and this ...
0
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0
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Why for a one-parameter subgroup $\phi$ of $G$ there is a vector field $X$ such that $\frac{d\phi^{\mu} (t)}{dt}=X^{\mu}(\phi (t))$?
Let $G$ be a Lie group and $\phi:\mathbb{R}\to G$ a one-parameter subgroup of $G$; i.e., $\phi (t)\phi (s)=\phi (t+s)$. Given a one-parameter $\phi$, why there is a vector field $X$ such that $\frac{d\...
1
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1
answer
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Why ins't this SU(4) Matrix produced by the exponential map?
I was working with the SU(4) Lie group, which is compact and simply connected. This should imply that the exponential map is sujective on the group.
However i came across the matrix $$G=\begin{pmatrix}...
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Construction of spin ½ representations of SO(p,q) from clifford algebras
There is a way of constructing spin ½ representations of SO(1,3) directly from the Clifford algebra $ \mathcal{C}\ell (1, 3) $ which is:
1 take a spin ½ representation of the Clifford algebra, for ...
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1
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Constructing the finite-dimensional representations of $\frak{sl}_3$
The finite-dimensional representations of $\frak{sl}_2$ can be constructed explicitly as partial differentiation of polynomials in two variables. Does a similar explicit construction exist for the ...
2
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4
answers
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Small question regarding left invariant vectorfields
Usually, one writes out the left invariance of vector fields $X$ on a Lie group $G$ as $$(L_x)_*X=X$$for every $x$.
However, I have trouble understanding this equality, since both sides does not map ...
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1
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a group isomorphic to $O(2N)$
Consider the set of $2N\times 2N $ complex matrices $T$ satisfying the conditions
$$TT^\dagger = I_{2N}$$
and ($^*$ means taking the complex conjugate)
$$ T^* = \gamma T \gamma ,$$
where $\gamma $ is ...
4
votes
1
answer
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Solving $\mathrm{e}^{\partial_x}u(x)=f(x)$
\begin{align}
\mathrm{e}^{\partial_x} u(x)=f(x),\quad x\geq 0
\end{align}
The formal solution is $u(x)=\mathrm{e}^{-\partial_x}f(x)$
the fact $[ {-\partial_x}, {\partial_x}]=0$ (Lie bracket) implies ...
5
votes
1
answer
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The Taylor series for product of Lie group elements
Let $x$ and $y$ be two elements of a Lie group $G$. In chapter 2 of the text "Lie Groups and Lie Algebras I" by A. L. Onishchik, the author states that, if $\overline{x}$ and $\overline{y}$ denote the ...
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Show that if $G$ is a Lie group with a bi-invariant metric, the left- and right-invariant vector fields of $G$ are Killing fields.
How do I show that if $G$ is a Lie group with a bi-invariant metric, the left- and right-invariant vector fields of $G$ are Killing fields?
Here is my attempt. If $X$ is a left- or right-invariant ...
2
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1
answer
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integral curves and Lie group actions
given a vector field $ X $ on the tangent bundle $ T\mathcal{M} $ to a manifold $ \mathcal{M} $ the flow of the field $ X $ is the map $ \Phi_X : \mathbb{R} \times \mathcal{M} \rightarrow \mathcal{M} $...
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Are $SL(n, \mathbb{C})$ and $U(n)$ transverse at the identity as submanifolds of $GL(n, \mathbb{C})$?
We know that if two embedded submanifolds $S_1$ and $S_2$ transversely intersect, then $T_p(S_1 \cap S_2) = T_p(S_1) \cap T_p(S_2)$. On the other hand, we have that $\operatorname{Lie}(SU(n)) \cong T_{...
1
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1
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Generators of Lie Algebra
The following comes from M. Maggiore, "A Modern Introduction to Quantum Field Theory". The author introduces a representation $D_R(g)$ of a group element $g=g(\theta)$ depending on the set ...
35
votes
2
answers
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Geometric intuition behind the Lie bracket of vector fields
I understand the definition of the Lie bracket and I know how to compute it in local coordinates.
But is there a way to "guess" what is the Lie bracket of two vector fields ? What is the geometric ...
8
votes
1
answer
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Is requiring diagonalizable adjoints vacuous in the definition of Cartan subalgebra?
One of the definitions of Cartan subalgebra $\mathfrak{h}$ of a semisimple Lie algebra $\mathfrak{g}$ one can find in the literature is that it is a
maximal abelian subalgebra
has the property that $...
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0
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The sum of two Maurer-Cartan forms is a Maurer-Cartan form
Let $\omega_1, \omega_2$ be two Maurer-Cartan forms, that is 1-forms satisfying the Maurer-Cartan equation
$$d\omega_i+\frac{1}{2}[\omega_i, \omega_i]=0$$
for $i=1, 2$. Then, it seems that $\omega_1+\...
0
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0
answers
43
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When is $\mathrm{exp}: \mathfrak g \rightarrow G$ surjective? [duplicate]
Let $G$ be a connected Lie group. What other conditions on $G$ are necessary and sufficient for the exponential map
$$\mathrm{exp}: \mathfrak g \rightarrow G$$
from the Lie algebra $\mathfrak g$ of $G$...
0
votes
1
answer
48
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"Boundary" of a Lie group
From my understanding of Lie groups, they can have multiple disconnected components. For example $\operatorname{GL}(n,\mathbb{R})$ has 2 components which can be seen since $\det$ is a Lie group ...
0
votes
1
answer
100
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How to define $SO(3)$ natural topology
Starting on some links on MSE -- e.g. $RP^3\cong SO(3)$ it is clear to me the intuitive reason why $SO(3)$ is identified with $RP^3$.
From a formal point of view, how is $SO(3)$ topologized ? Can we ...
2
votes
2
answers
299
views
Derivative of left-multiplication in $GL(n,\mathbb{R})$
Given two maps between sets $f \colon X \to Y$ and $g \colon Y \to Z$, we define the pullback of $g$ by $f$ as $f^*g := g \circ f$.
Given a smooth map between manifolds $F \colon M \to N$ of ...
1
vote
0
answers
76
views
If $G$ is a simply connected Lie group and $H^2 (\mathfrak{g};\mathbb{R})=0$, then $H^2 (G;\mathbb{R})=0$?
Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra.
My question is that: If $G$ is a simply connected and $H^2 (\mathfrak{g};\mathbb{R})=0$, then $H^2 (G;\mathbb{R})=0$?
My question ...
0
votes
1
answer
22
views
Central extensions versus 2-cycles
Let $G$ be a Lie group and $[C]\in H^2 (G;\mathbb{R})$. Then $\hat{G}_C :=G\times \mathbb{R}$ with multiplication $(g,r)\cdot (f,s):=(gf,r+s+C(g,f))$ (i.e. $\hat{G}_C=G\ltimes \mathbb{R}$) is a ...
1
vote
1
answer
64
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If we have two Dynkin diagrams such that $D_{1}$ is a subdiagram of $D_{2}$. Then the $\mathfrak{g}(R_{1})$ is a subalgebra of $\mathfrak{g}(R_{2})$
Recently I was studying my notes on Lie algebra, and while I was studying Dynkin diagrams, I came with the following question:
If we have $D_{1}$ and $D_{2}$ two Dynkin diagrams such that $D_{1}$ is ...
2
votes
2
answers
132
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Are "infinitesimal rotations" commutative? If so, which mathematical fact allows it?
I was reading Moysés Nussenzveig's "Basic Physics Course 1" when I came across this excerpt in chapter 11, about rotations and angular momentum, in section 11.2, vector representation of ...
3
votes
1
answer
70
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What is in the image of the exponential of $\frak{sl}(n,\mathbb{R}$)? What do you need to get all of $\mathrm{SL}(n,\mathbb{R}$)?
This question discusses how $\mathrm{S}L(2,\mathbb{R}$) coincides with $\pm\exp(z)$ with $z\in \frak{sl}(n,\mathbb{R}$) (the real traceless matrices). Is it known what happens for $n>2$?
Namely, ...