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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.

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Topology problem on Lie Transformation groups.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and at the page 14 is this lemma: My question is why the uniqueness of that topology is trivial? I just know that i have ...
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Continuous automorphism of a lie group in kobayashi's book

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and i dont understand a thing at page 14. My question is why $A_\varphi$ is continuous? $G$ is a subgroup of ...
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For $GL(n, \mathbb{R})$, how do I show that multiplication and inverses are smooth maps?

I'm learning about Lie groups, and $GL(n, \mathbb{R})$ seems to be one of the first non-trivial examples. I know that this is a group and it is a smooth manifold, but I don't understand how to view ...
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21 views

Showing the following map is surjective

I am trying to use the implicit function theorem to prove that $Sp(4,\mathbb{R})$ is a Lie group. I have defined the map $f:M_{4\times 4}(\mathbb{R}) \to \mathit{Skew}_{\mkern 1.5mu 4\times4}$ by $f(X)...
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26 views

Complex structure on lie algebra?

This is related to an Lemma 7.34 of Brian Hall, Lie group, Lie Algebra and Representations chpt 7. Lemma 7.34: Let $K$ be a compact matrix lie group with non-commutative lie algebra. Then real lie ...
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factoring a neighborhood of identity in a compact connected Lie group with a closed Lie subgroup

Given a closed Lie subgroup of a compact and connected Lie group, it seems plausible to me that there exists such a neighborhood of identity in the original Lie group that it can be written as the ...
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29 views

Is the image of a matrix Lie group under a Lie homomorphism again a Lie group?

Suppose $G\in GL_{n_1}(\mathbb C),H\in GL_{n_2}(\mathbb C)$ are matrix Lie groups such that $\theta:G\to H$ is a Lie homomorphism . Then is the image of G under the map necessarily a matrix Lie group? ...
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$\{(X,Y)\in \Bbb R^3: |X|=|Y|=1, \langle X,Y \rangle=0\}$ is homeomorphic to $\text{SO}(3, \Bbb R)$

Let $R=\{(X,Y)\in \Bbb R^3: |X|=|Y|=1, \langle X,Y \rangle=0\}$ with $\langle\,\cdot\,,\,\cdot\,\rangle$ the Euclidean scalar product. Prove that $R$ is homeomorphic to $\text{SO}(3, \Bbb R)$. I ...
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Characteristic classes for $P \rightarrow G \rightarrow G/P$

Let $G$ be a complex semisimple Lie group and let $P$ be a parabolic subgroup. We know that the cohomology of the flag variety $G/P$ is generated by Schubert classes. There is a principal $P$ bundle, ...
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Help find mistake in conclusion about vector fields on principal bundle

Let $P$ be a principal fiber bundle with structure group $G$ acting freely on the right. Let $T_uP = G_u + Q_u$ be a connection on $P$, where $u\in P$ and $G_u$ is the vertical space consisting of ...
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26 views

Show Sp($n$)$\subset$SU($2n$)

On the Wikipedia page on symplectic groups, it is stated that Sp($n$)$\subset$SU($2n$). How can this be shown?
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Showing surjectivity of exponential map.

TL;DR: How to show eq. (2) using eq. (1)? I'm currently having a bit of a hard time with the following problem. In class we showed that $$\mathfrak{so}(1,3;\mathbb{C})\cong \mathfrak{sl}(2,\mathbb{C}...
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Non-singular square matrix with unit (but not necessarily mutually orthogonal) column vectors

Consider the general Linear group $GL_{\mathbb R}(n)$ quotiented (on the right) by the subgroup $(\mathbb R^\times)^n$: $$(\mathbb R^\times)^n:=\{\text{diag}(\lambda_1,\dots,\lambda_n)~|~\lambda_i\in\...
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33 views

Left-invariant vector field is differentiable

I am quite new to the subject, please comment if something does not make sense. Let $M$ be a Lie group. $L_x$ be left multiplication by $x\in M$. I want to show that vector space $X:M \to TM$ ...
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Abstract Weyl group for a Lie algebra

Can someone please help me with how I find the Weyl group (abstractly) from the Cartan Matrix. I am using the $A_2$ root system as my example and so far have the Coxeter matrix for that but unsure ...
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Finding the Weyl group from a Cartan matrix

I am looking to establish the relationship between Weyl groups and semisimple Lie algebras. So far I have found the root space decomposition (I am using $\mathfrak{sl}(3, \mathbb{C}) $as my example). ...
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Understanding how the Lie algebra G2 arises in nature

I'm trying to understand these notes on G2. But I don't understand the very beginning. I have no background in physics at all. What is meant by a "configuration space?" I looked on wikipedia but ...
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Unitary representation of the Heisenberg group and the universal enveloping algebra

I am studying the Heisenberg group with the Lie algebra generators $\{ U,V,W \}$ and the structure $[U,W]=[V,W]=0$ and $[U,V]=W$. This group has an infinite-dimensional unitary representation on the ...
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29 views

Necessity of diagonalizability of adjoint representation of cartan subalgebra in definition

This is the definition of cartan subalgebra define in Brian Hall, Lie groups, Lie algebras and representations, 2nd Edition, Chpt 7, Sec 2, Def. 7.10. I am assuming the ground field is $C$ or it does ...
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14 views

Coset decomposition of a Lie algebra with a compact symmetric subalgebra

If the Lie algebra $\mathbf G$ is connected and $\mathbf G = \mathbf K \oplus \mathbf P$ where $\mathbf K$ is a compact symmetric subalgebra, $$ [\mathbf K, \mathbf K]\subset \mathbf K, \ [\mathbf ...
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General form of elements of $SU(2)$

$SU(2)$ is the set of $2\times 2$ complex matrices $A$ satisfying $AA^*=I$ and $\det(A)=1$ where $A^*$ denotes the conjugate transpose of $A$ and $I$ is the identity matrix. I've seen everywhere that ...
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Lie algebra: intuition of “Lie Algebra is tangent space of corresponding Lie Group”?

I am an engineering student and learned of Lie Group/Lie Algebra recently. I can follow and understand all the formula derivation of Lie Algebra from Lie Group. But I cannot grasp the meaning of "...
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An Exercise about Riemann Geometry

Let G be a compact Lie group with a bi-invariant metric. (a)Let p be a point, and let q be conjugate to p along a geodesic . Show that the dimension of the space of Jacobi fields along vanishing ...
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There exist only finitely many λ ∈ C such that det(λA + (1 − λ)B) = 0

I found this question in the Lie Groups book, this is the start of the proof of connectedness of GL(n, C). A, B ∈ GL(n, C), show that there exist only finitely many λ ∈ C such that det(λA + (1 − λ)B) ...
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Cotangent lift of left translation SE(3)

I am wondering how I can verify the cotangent lift of left translation $ T^*L_{(\Lambda,\phi)}:T^*_{(\Lambda,\phi)}G\to T^*_eG$ which reads $$ (\Lambda,\phi)^{-1}(\alpha_\Lambda,(\phi,v))= T^*L_{(\...
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3answers
55 views

When does $P\text{SO}P^{-1} \subseteq \text{SO}$?

Let $P \in \text{GL}_n^{+}(\mathbb {R})$. Suppose that $P\cdot \text{SO}(n)\cdot P^{-1} \subseteq \text{SO}(n)$. Is it true that $P \in \lambda \text{SO}(n)$ for some $\lambda \in \mathbb{R}$? I ...
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Can the formula for the derivative of the exponential map (in a Lie group) be derived without using the power series formula?

I've been reading about Lie Groups/Algebras recently, and most proofs I've seen for derivative of the exponential map involve expressing it as a power series at some point. I'm curious if there's a ...
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35 views

Adjoint action on universal enveloping algebra

Let $G$ be a Lie group, $\mathfrak{g} = \operatorname{Lie}(G)$ be a Lie algebra and $U\mathfrak{g}$ be the universal enveloping algebra of $\mathfrak{g}$. I want to show that if $D\in Z(U\mathfrak{g})$...
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Dominance of two real-valued funstions on manifold

I have two $\mathbb{R}$-valued functions $f,g : G \to \mathbb{R}$ on a Lie group $G$. For example, $G = SO(3)$. The two functions $f,g$ are ugly and it is hard to compute the exact value of $f(M), g(M)...
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$G$ is Lie group and $V$ is a representations of $G$,prove representations $V \otimes V \cong S^2(V) \oplus \Lambda^2(V)$

Let $G$ a Lie group and let $V$ a representations of $G$. Then we have the following representations are isomorphic: \begin{align} V \otimes V \cong S^2(V) \oplus \Lambda^2(V) \end{align} I have no ...
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Separating the compact part of a Lie group

I am interested in knowing which class of Lie groups $G$ can be decomposed as a semi-direct product $$ G \sim N \rtimes H$$ where $N$ is a Lie group whose exponential is injective, and $H$ is a ...
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interaction of left-invariant vector fields and right-translation on a Lie-Group

Given a Lie-Group $G$ denote the set of left-invariant vector fields on $G$ by $LG$ and denote by $R_g$ the right-translation, i. e. for $g \in G$ define $$R_g \colon C^\infty (G) \to C^\infty (M) \...
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Is $GL_n(\mathbb{C})$ under trace norm homeomorphic to some $\mathbb R^m$?

I've often seen proofs that invoke the Heine-Borel theorem to show that certain matrix groups are compact. They view the matrix groups (such as $SO(n)$ as $\mathbb{R}^{2n}$ and then go on to prove ...
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Lie algebras of the unit group of a finite dimensional algebra

Given a finite dimmensional associative algebra $A$, it can be proven that the unit group $$A^{\ast}:=\{a\in A \mid \exists\, b \in A \mbox{ such that } ab=ba=1_A\} $$ is always a Lie group (The unit ...
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Definition of $\Gamma$-invariant set with $\Gamma \subset O(n)$

I am reading a paper, the first sentence of "Preliminaries and notations" says We assume $\Gamma$ to be a finite subgroup of $O(n)$ and $X\subset \mathbb{R}^n$ to be a compact $\Gamma$-invariant ...
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classical mechanics in coadjoint orbits

We know that coadjoint orbits are symplectic manifolds, and they can be used to find unitary representations of lie groups and stuff, and it's also related to quantization. However, is it true that ...
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Strong Approximation theorems in Function Fields versus Groups

In the context of global function fields, the strong approximation theorem can be stated as follows: Let $F$ be a global function field, and $P_1,P_2,\dots,P_r$ be a finite set of places of $F$ (with ...
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1answer
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Can every positive root of a Coxeter group be written as a simple root and a positive root?

Can every positive root of a Coxeter group be written as a simple root and a positive root? I think that this is possible. For example, in type $B_2$, the set of positive roots are $\alpha_1, \alpha_2,...
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1answer
66 views

Can we study representation of $p$-adic group by studying $p$-adic Lie algebra?

While I'm studying about representation theory of $\mathrm{GL}(2)$ over local fields, I found that there's no one talking about $p$-adic Lie algebra. However, for Lie groups over $\mathbb{R}$ or $\...
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Center of the universal enveloping algebras as the space of $\operatorname{Ad}$-invariant elements

Let $G$ be a smooth Lie group with Lie algebra $\mathfrak g$. The adjoint action of $G$ on $\mathfrak g$ extends to the complexification $\mathfrak g_{\mathbb C}$ and then to the universal enveloping ...
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Counterexample to Lie's second theorem for SO(3)

Lie's second theorem says that if $G$ is a simply connected Lie group, then every isomorphism $\Phi$ of its Lie algebra $\mathfrak{g}$ lifts to an isomorphism $\phi$ of $G$, i.e. such that $d\phi_e = \...
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Understanding proof that exponential map of compact connected Lie group is surjective

Let $G$ a compact connected Lie group. Then, the exponential map $\exp: LG \rightarrow G$ is surjective. (where $LG$ is the Lie Algebra of $G$). $\textbf{Proof:}$ For any torus $T' \subset G$ we have ...
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Does there exist any non-trivial linear relation on the components of elements of $O(2,1)$?

Consider as an example $$O(2)=\left\{\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\middle|\ \ \theta\in \Bbb R\right\},$$ clearly for $g\in O(2)$ one has $g_{...
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1answer
31 views

Existence of diagonal matrix that transforms invertible matrix in unitary

This questions arises from the answer to this previous question, which leaded me to 'relax' the statement that I wanted to prove. Suppose that a matrix $M\in\mathrm{SL}_2(\mathbb{C})\setminus\{\pm I\}...
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1answer
71 views

Euler characteristic expression in terms the number of fixed points of an $\mathbb{S}^1$ action

I have found in a paper* that I am reading that Given $(M,J)$ compact (smooth) manifold with an almost complex structure $J$, if we have an $\mathbb{S}^1$ action with isolated fixed points then $ \...
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2answers
45 views

$d(exp)_{0}:T_0 \frak{g} \to$ $T_eG$ is the identity map

I'm learning about Lie groups, and do not have a thorough background in differentiable manifolds. I have the following definition: For a map $F:M \to N$ between manifolds and $a \in M$, the ...
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1answer
21 views

Prove commutation with tensor product

Say I have $\left|1,\psi\right>$ which is a vector in one representation of $\mathcal L$, where $\mathcal L$ is a Lie algebra and $\left|2,\phi\right>$ which is a vector in another ...
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32 views

Lie algebra definition problem.

The question is: How do we define the Lie algebra generated by a set of vector fields? I'm reading Kobayashi's book Transformation Groups in Differential Geometry and in the proof of the next theorem ...
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1answer
27 views

Action of a Closed Lie subgroup

Suppose $G$ is a Lie group acting properly on a manifold $P$. Does it imply the action of any closed subgroup $H$ on $P$ is also proper?
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The decomposition into two irreducible SU(4) representations

I read in the book "String Theory and $M$-theory: A Modern Introduction" that the eight components can be decomposed into two irreducible $SU(4)$ representations: $8=4 \otimes \bar{4}$. Is it correct ...