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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What are examples of nilpotent Lie groups except abelian and matrix groups?

What are some notable examples of nilpotent Lie groups, excluding abelian groups and matrix groups, along with their associated Lie algebras? Are there any classifications of such groups
NIshant Rathee's user avatar
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Universal enveloping algebra from analytical point of view

Can someone fully explain the universal enveloping algebra from an analytical point of view? I would like to see it first defined in terms of distribution, then in terms of left invariant ...
Lefevres's user avatar
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Interpolation of perturbed rotations and approximating the linearized effect of doubly perturbed rotations

I am reading through "State Estimation for Robotics" by Timothy Barfoot and I came across a line that I don't understand in pg 242, equation (7.136) Suppose we have the following definitions:...
humble_torch_student's user avatar
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Decomposition of Adjoint Group of Semisimple Lie Group

I have trouble understanding the beginning sentence of section $3$ of a paper by C.Moore: https://www.jstor.org/stable/2373052, which I quoted below: Suppose now that $G$ is a (connected) semi-simple ...
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The mathematical core of the Spin-Statistics Theorem

Presently I am trying to understand the mathematics behind the spin-statistics theorem from quantum theory (fermions' 1/2 integer spin implies antisymmetric states, bosons' integer spin implies ...
hatodd's user avatar
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Why is $O(n)$ not the double cover of $SO(n)$?

$O(n)$ has two connected components, $(\det)^{-1}(1)$ and $(\det)^{-1}(-1)$. While I know it is not true, I am wondering why the above is not enough to say that $O(n)$ is the double cover of $SO(n)$ ...
CBBAM's user avatar
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Why don't I see any mention of fundamental domain in the context of non-discrete Lie Group acting on Riemannian manifolds

Let $G$ be a non-discrete Lie Group acting properly on a Riemannian manifold $M$ by its isometries. When $G$ is discrete, there are plenty of literature on Fundamendal Domains (FD), but a quick ...
Learning Math's user avatar
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An attempt of solving the tangent map of left multiplication by $g$ on $GL(n,\mathbb{R})$ is also a left multiplication

This problem is from "An Introduction to Manifolds" by Loring W. Tu. I think I understand the solution, but I would like to ask if the following idea of approaching it is right. When I dealt ...
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Center of Levi subgroups in SLn

Fix a partition $\lambda$ of $n$ of length $l$. Consider a Levi subgroup $L_{\lambda}$ in $GL_{n}(\mathbb{C})$. There is the decomposition $L_{\lambda} = \prod_{i} GL_{\lambda_{i}}$. Now the ...
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Given a simple Lie group $G$ with several subgroups $S_n$. Under what conditions do they generate $G$? [closed]

This question arises from my studying on linear algebra and group theory. Suppose I only have several matrices, is it possible to construct every possible matrix by cascading (=matrix multiplication) ...
XjX's user avatar
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Why is the group of rotations in $\mathbb{R}^n$ not $n$-dimensional?

I have a rather basic mis-understanding about Lie groups and Lie algebras. Consider the Lie group $SO(N)$ for $N>3$ of rotations on $\mathbb{R}^N$. On the one hand this Lie group has dimension $N(N-...
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Invariant vector field (under translation) are constants.

Example 8.40 (Introduction to smooth manifolds. Lee) If we consider $\mathbb{R}^n$ as a Lie group under addition, left translation by an element $b\in\mathbb{R}^n$ is given by the affine map $L_b(x)=b+...
eraldcoil's user avatar
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When does A and exp(B) commuting imply A commutes with B?

Let $A,B$ $\in GL_{n}(\mathbb{C})$ and $[A,B] = AB-BA = 0$. My question is about the existence of a $b \in M_{n}(\mathbb{C})$ such that $B = \exp(b) $ and $[A,b] =0 $. Note that in general $[A,\exp(b)...
arczn's user avatar
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Verifying that a vector is a highest weight vector

I'm trying to show that $(e_1 \otimes e_2 \otimes e_3 - e_3 \otimes e_1 \otimes e_2) \otimes e_n^*$ is a highest weight vector for the irreducible submodule of $V^{\otimes 3} \otimes V^*$ with highest ...
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What's ${\cal su}(2)_{2n}$? [closed]

I saw a notation where a subscript was given to a lie group notation ${\cal su}(2)_{2n}$. What's ${\cal su}(2)_{2n}$?
ShoutOutAndCalculate's user avatar
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Tangent Space of Torus

Let T be a torus (namely, topology homeomorphic to quotient topology $\mathbb{R}/\mathbb{Z}$). Intuitively it makes sense to say that tangent space of torus at identity (i.e. lie algebra of 1D torus) ...
엄익훈's user avatar
7 votes
2 answers
403 views

Smallest group acting transitively on projective space

Let $ K $ be a field. Let $ K^n $ be an $ n $ dimensional vector space over $ K $. Let $ KP^{n-1} $ be the projective space of lines in $ K^n $. Let $ GL(n,K) $ be the group of invertible $ n \times n ...
Ian Gershon Teixeira's user avatar
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1 answer
108 views

Is $SL(2,\Bbb R)$ generated by $SO(2)$ and a single upper triangular element?

Consider the subgroup $\Gamma < \operatorname{SL}(2,\mathbb{R})$ generated by the element $$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} $$ and all elements of $\operatorname{SO}(2)$. Is $\...
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Identity regarding the differential of a smooth action of a Lie group

Hey I am currently following a course on Lie groups and I have followed a course on smooth manifolds. The question I have is the following, Let $G$ be a Lie group acting smoothly on a manifold $M$ by ...
Ronnie's user avatar
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A confusion about Lie group of two dim non abelian real Lie algebra

In the book by Fulton & Harris (chapter 10, section 1) : Finally, in the real case things are simpler: when we exponentiate the adjoint representation as above, the Lie group we arrive at is ...
Eloon_Mask_P's user avatar
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How to define periodicity of orbits for general group actions?

Let $H$ be a topological group acting on a topological space $X$. Is there a general definition of periodicity in this case? Write $X=G/\Gamma$ and consider the orbit $Hg\Gamma$, what does it mean for ...
taylor's user avatar
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Proof that $O_3$ and $GL_3$ are homotopy equivalent

Proof that $O_3$ and $GL_3$ are homotopy equivalent Trying to use polar decomposition to solve this problem and actually I have come to the fact that exists continuous function $sqrt(AA^T) = C$ where ...
yehehhd's user avatar
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What is the definition of an affine $G$-module?

Let $G$ be a Lie group, $T:G\to GL(\mathbb{V})$ a representation of $G$ in a vector space $\mathbb{V}$. A $\mathbb{V}$-valued one-cocycle is a (smooth) map $S:G\to V$ satisfying the property $S(fg)=T(...
Mahtab's user avatar
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Why is the complex Lie group $(\mathbb C^*)^n$ called "Complex Torus"

While studying complex Lie groups theory, and more generally complex geometry, I've found two different objects which are called "complex tori". Consider the multiplicative group $\mathbb C^...
Federico T.'s user avatar
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1 answer
66 views

Double cover vs universal cover for ${\rm SO}^+(p,q)$

I am confused about the double covers and universal cover of ${\rm SO}^+(p,q)$, where by this notation I mean the connected component with the identity. Previously I thought that ${\rm SO}^+(p,q)$ had ...
Gold's user avatar
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Why is the diffeomorphism group a manifold?

Let $M$ be a differentiable manifold. The diffeomorphism group of $M$ is the group of all $C^{\infty}$ diffeomorphisms of $M$ to itself, denoted by $\text{Diff}(M)$. This space of diffeomorphism $\...
Swakshar Deb's user avatar
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Proving a Maurer-Cartan type equation without differential forms

I found this very nice theorem in the book by Onishchik ,"Lie groups and Lie algebras I". Proposition 2.9. Let $g(t,s)\in G$ be a smooth map from some domain in $\mathbb{R}^2$ into a Lie ...
level1807's user avatar
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How we define the Lie symmetry of a stochastic differential equation?

In the literature of symmetry groups, Sophus Lie define the symmetry of a pde or an ode by a vector field defined in the tangent space of the submanifold (defined by solutions of the pde or ode) and ...
Anas Cobain's user avatar
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When is the quotient of $\operatorname{GL}_n(\mathbb{R})$ by a discrete subgroup compact?

My question is exactly that on the title. I'm interested in the action of some (discrete) subgroup $H$ on $\operatorname{GL}_n(\mathbb{R})$ by left multiplication. For example, $H$ can be $\...
Bumblebee's user avatar
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What does "extend the identity function" mean?

In Baker A.'s book section 3.6 about the complexification of a real lie algebra, it briefly introduces one theorem: If $\mathfrak{g'}$ and $\mathfrak{g''}$ are two complexifications of $\mathfrak{g}$ ...
Krystal Justin's user avatar
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Why or how to prove that the infinitesimal symmetries of a differential equation form a Lie algebra?

Given a partial differential equation, after computing the Lie point symmetries of this PDE, means a one-parameter group (Lie group) of transformation that leaves the set of solutions of the PDE ...
Anas Cobain's user avatar
1 vote
1 answer
52 views

Does the inner semidirect product of Lie groups need these two subgroups both be closed?

I am studying GTM218 and found an unproven theorem that the author left for exercise. Here is the theorem: Theorem 7.35 (Characterization of Semidirect Products). Suppose $G$ is a Lie group, and $N;H \...
SiberiaCat's user avatar
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1 answer
75 views

If $G$ is an abelian Lie group then the Lie algebra of $G$ is abelian

This is problem 8-25 from John Lee's Introduction to Smooth Manifolds. Prove that if $G$ is an abelian Lie group, then $Lie(G)$, the Lie algebra of the Lie group $G$ is abelian. [Hint: show that the ...
nomadicmathematician's user avatar
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Fundamental Group of Compact Connected Lie Group

A compact, connected Lie group is isomorphic to the quotient of a torus $T$ and a simply-connected, compact Lie group $J$ by a finite central subgroup $Z\subseteq T\times J$: $$G\simeq (T\times J)/Z.$$...
Chistlo's user avatar
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1 answer
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Explicit formula for the action of $\mathrm{SL}(2,\mathbb R)$ on the Poincaré disk

After googling a bit I was surprised that I couldn't find any reference in which the action of $\mathrm{SL}(2,\mathbb R)$ on the Poincaré disk corresponding to the usual action by Möbius ...
B K's user avatar
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Explicit expression of matrix $e^A$ where $A\in\mathfrak{so}(1,3) $

Let $\eta=$ diag$\,(-1,1,1,1) $. Consider the proper orthochronous Lorentz subgroup $SO^+(1,3)$ which contains all matrix $L=(a_{ij})_{0\leq i,j\leq3}$ such that $L^\top\eta\,L=\eta $, $\det L=1$ and ...
PermQi's user avatar
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1 answer
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Relating integration of forms to Haar measure integration on a Lie group

Let $G$ be a Lie group with Haar measure $\mu$ and Lie algebra $\mathfrak{g}$ given as the space of left-invariant vector fields on $G$. I want to understand the relationship between integration of ...
user920957's user avatar
1 vote
1 answer
64 views

Action of $E_{ij}$ matrix on vector in $V^{\otimes 3} \otimes V^*$

I'm trying to understand a calculation in a paper in which a composition of $\text{GL}_n(\mathbb{C})$ maps are applied to a vector in $\wedge^2 V \otimes V^*$ (where $V$ is the standard $\text{GL}_n(\...
Chase's user avatar
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Question regarding the exponential property of an integral curve and the corresponding one parameter group.

I am basically a beginner in the manifold course. We have seen that if $X$ is a vector field given over a manifold $M$ and $x\in M$ then there exists a curve passing through $x$ such that locally the ...
Kishalay Sarkar's user avatar
3 votes
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An example of non simple group which is also a Lie group such that $G$ is a connected Lie group and has no non trivial normal Lie subgroup

I want to know if there exist a non simple group (as abstractl group) $G$ such that $G$ is a connected Lie group and has no non trivial normal Lie subgroup. I have tried some obvious examples like ...
Eloon_Mask_P's user avatar
7 votes
0 answers
116 views

Is the monster group maximal in SO(196883)?

The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the orthogonal group. In fact, since ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
31 views

Why is the stabilizer of a holomorphic complex Lie group action a complex Lie subgroup?

Suppose $G$ is a complex Lie group and $M$ is a complex manifold. Suppose we have an action of $G$ on $M$ which is a holomorphic map $G \times M \rightarrow M$. I have seen the claim that it is easy ...
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Image of exponential map for lie group $SL(2, \mathbb{R})$

Currently I am studying lie theory and differentiable manifolds together, and I encountered a question regarding the image of exponential map from lie algebra of $SL(2, \mathbb{R})$ on to $SL(2, \...
엄익훈's user avatar
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1 answer
126 views

The group $SO(1,3)$ and its Lie algebra

Denote the matrix $\eta=$ diag$(-1,1,1,1)$. The group $O(1,3)$, called Lorentz group, is the group of all matrix $L\in M_4(\mathbb R)$ such that \begin{align} L^\top\eta\,L\,=\,\eta.\tag1 \end{align} ...
PermQi's user avatar
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0 answers
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Universal enveloping algebra as the algebra of left-invariant differential operators

I have come across the following statement multiple times: Let $G$ be a Lie group. The universal enveloping algebra of $G$ can be identified with the algebra of left-invariant differential operators ...
mixotrov's user avatar
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A question on alternating product and $SL_n(R)$

I am reading Fulton and Harris representation theory. In the section 8.2 - Examples of Lie Algebras, while calculating the Lie algebra of $SL_n(\mathbb R),$ the author tell that by definition $$ A_t(...
Eloon_Mask_P's user avatar
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12 views

morphisms between stable unitary, orthogonal, and (compact) symplectic groups

The subgroup inclusions $$ U(N) \hookrightarrow Sp(N):=U(2N) \cap Sp(2N,\mathbb{C}), \quad U(N) \hookrightarrow O(2N) $$ induces some morphisms $$ f_1: U(\infty)\to Sp(\infty), f_2: U(\infty)\to O(\...
Hyeongmuk LIM's user avatar
2 votes
0 answers
52 views

The interpretation of Jacobi identity in terms of Lie group

Lie algebra and Lie groups are closely related for example by exponentiation map. Given the commutators of Lie algebra generators, one can compute the multiplications between Lie group elements by ...
user39511's user avatar
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Nilpotent subalgebras and ideals

I believe these questions are trivial, but I still need some help. Consider $\mathfrak{h}$ Lie subalgebra of the real $n-$dimensional Lie algebra $\mathfrak{g}$. Is there a non-trivial ideal $J$ of $\...
Tmath's user avatar
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Chern-Weil theory in terms of pullback along classifying map

Let $G$ be a semisimple real Lie group and $M$ be a smooth manifold. The map on cohomology $H^{\ast}(BG,\mathbb{C}) \rightarrow H^{\ast}(BT,\mathbb{C})\simeq \mathbb{C}[\mathfrak{h}]$ induced by the ...
zygomatic's user avatar

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