# 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
1 vote
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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 ...
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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:...
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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 ...
1 vote
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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 $SO(n)$ ...
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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 ...
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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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1 vote
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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) ...
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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 ...
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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 ...
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1 vote
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### 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 ...
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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 $Z\subseteq T\times J$: $$G\simeq (T\times J)/Z.$$...
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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 ...
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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 ...
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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 ...
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### 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} ...
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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 ...
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