Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
3,059 questions
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Lie algebras and complements
I have some elementary questions about Lie algebras and vector space complements.
Let $(\mathfrak{g},[.,.])$ be a finite-dimensional Lie algebra and $\mathfrak{g}_1$ a Lie ideal in $\mathfrak{g}$.
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Lie group actions and f-relatedness
Background
Let $f: M \to N$ be a smooth map between smooth manifolds.
Two vector fields $X$ in $M$ and $Y$ in $N$ are said to be $f$-related if for all $p \in M$, $(f_*)_p(X_p) = Y_{f(p)}$; ...
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The lie algebra of the orthogonal group of an arbitrary space time metric
Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...
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What is the situation with Hilbert's Fifth Problem?
The common knowledge in this regard seems to be that Hilbert's Fifth Problem was completely solved by Gleason, Montgomery, and Zippin. However, such wisdom was contested by Peter Olver:
Olver, Peter ...
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What are the computationally useful ways of thinking about Killing fields?
One definition of the Killing field is as those vector fields along which the Lie Derivative of the metric vanishes. But for very many calculation purposes the useful way to think of them when dealing ...
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Is there a way to see a topological group as the "Cayley graph" of its "infinitesimal generators"?
At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...
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The orthogonal group of a riemannian metric
Let the inner product of the vectors X and Y on a given four dimensional manifold (EDIT: make this R4) be defined as (X*Y) = gikXiYk; using the summation convention for repeated indicies.
Let A be a ...
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Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
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Action of the group of isometries on a manifold
Hi guys,
I am able to prove that any symmetric manifold is complete (Consider a local geodesic and use the symmetry to flip it, effectively doubling the length of the geodesic, ad infinitum). I want ...
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Frobenius Theorem
Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$
Then I suppose the following properties hold for M,
There exists a metric ...
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Sections and subgroups in a unipotent group
Let $U$ be a smooth connected unipotent algebraic group over an algebraic closure $k$ of $\mathbf{F}_p$. Let $U' := [U,U]$ be the derived (= commutator) subgroup, and assume it is central in $U$. Let $...
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Is U(n) a normal subgroup of SO(2n)? [closed]
U(n) is the group of n by n unitary complex matrices and SO(2n) is the group of 2n by 2n real orthogonal matrices with determinant 1.So far I can show that how to get an injective group homomorphism ...
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How to Prove that nilpotent Lie groups satisfy the Leptin condition?
A topological group $G$ is said to satisfy the Leptin condition if for every compact subset $K\subseteq G$ and for every $\epsilon>0$ there exists a compact subset $L$ such that
$\mu(LK)$ < $ (...
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What matrix groups can be embedded in $Sp_4$?
In a joint paper with Yifan Yang we constructed an "exotic" embedding
of $SL_2(\mathbb R)$ in $Sp_4(\mathbb R)$ (in fact, of $PSL_2(\mathbb R)$ in $PSp_4(\mathbb R)$),
namely,
$$
\iota\colon\begin{...
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Semi-simple lie groups and their fundamental representations
I've got a really basic question on the representation theory of semi-simple Lie groups. I know that a rank-R semi-simple Lie group possesses R fundamental representations. But is the relation ...
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Jordan decomposition in a classical group
Let $\mathfrak{g} \subset \mathfrak{gl}_n$ be one of the classical real or complex semisimple Lie algebras. If $g \in \mathfrak{g}$, then $g$ has a Jordan decomposition $g = g_s + g_n$ with $g_s$ ...
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Is the quotient functor of points of a Lie group with the subfunctor of a closed subgroup a sheaf?
Let G be a Lie group and $H\subseteq G$ be a closed subgroup. It can be shown that $H$ has a unique differentiable structure such that the inclusion map $H\to G$ is an embedding of manifolds.
The ...
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How to show the matrix exponential is onto? And, how to create a powerseries for log that works outside B(I,1)
Hi,
I've been looking for a clear reference which shows that the matrix exponential is surjective from $M_{n}(C)$ to $Gl_{n}(C)$. Wikipedia claims this is true, but I haven't seen it proven... ...
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How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?
By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over ...
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The invariant 3-form on a compact Lie group
Let $G$ be a compact Lie group. We have the well-known Maurer-Cartan left-invariant and right-invariant $1$-forms $\theta$ and $\bar\theta$ in $\Omega^1(G, \mathfrak{g})$, probably discussed in every ...
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Adjoint orbits of small subspaces in Lie algebras
I've been studying isometric quotients (by compact Lie groups) of compact simply connected homogeneous spaces $G/H$ and their inherited curvature. One of the issues that continually arises is the ...
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Definitions of real reductive groups
There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind:
A closed subgroup of $GL(n,\mathbb C)$ closed under conjugate transpose.
The set ...
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Is there a Morse theory proof of the Bruhat decomposition?
Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...
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Defining a family of rotations with certain properties
Let $d \ge 2$, and consider the sphere $S^{d-1}$ embedded in $\mathbb R^d$. Does there exist a family of rotations $\{\mathcal O_v\}_{v \in S^{d-1}}$ which satisfies:
$\mathcal O_v e_1 = v$, and
$\...
- 8,512
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Infinite dimensional unitary representations of SU(2) for non-half-integer j?
The finite dimensional irreducible unitary representations of $SU(2)$ are labelled by $j$ which needs to be half-integer, the dimension of the representation is $2j+1$. This is well-known, all is good....
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Algebraicity of holomorphic representations of a semisimple complex linear algebraic group
Let $G$ be a complex linear algebraic group, given to us as a closed subgroup of some $\mathrm{GL}(n,\mathbb{C})$. Suppose moreover that $G$ is semisimple. Then it's a fact that every finite-...
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Equivariance of vector bundles over G/B
Let $G$ be a complex semisimple group, $B$ a Borel subgroup of $G$ and $X=G/B$ the flag variety of $G$. If $G$ is simply connected, then every line bundle $L$ on $X$ can be made $G$-equivariant (see ...
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The normalizer of a reductive subgroup
Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the ...
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How to calculate symmetric tensor products of SO(10) representations?
I just want to consider the simplest case:
Let S=[0,0,0,0,1], how to derive the general formula for $Sym^k$ S?
My conjectured formula based on the results of LiE program for finite k values is:
$...
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The Simply Connected Subgroups of GLn(C)?
A friend of mine and I were trying to answer a question related to his research and he couldn't remember whether or not the special linear group over the complex numbers, SLn(C),was simply connected. (...
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Questions on orbit properties of group action on varieties
Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...
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When is a conjugacy class of matrices an embedded submanifold?
Let $M_{n\times n}$ denote the set of $n\times n$ real matrices and let $GL_n$ be the subgroup of invertible matrices. $GL_n$ acts on $M_{n\times n}$ smoothly by conjugation, which means that each ...
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Terminology for nilpotent groups
I have a nilpotent lie group $N$ with upper central series
$$1 = N_0 \triangleleft N_1 \triangleleft \dots \triangleleft N_k = N$$
which induces the filtration $$0 = \mathfrak{n}_0 \subset \mathfrak{n}...
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sub-tori of a torus, generated by 1-dimensional subgroup
Ok the question is pretty dumb: suppose you have a torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$ and a vector $\bar{v}=(v_1,\ldots,v_n)\in\mathbb{R}^n$.
Consider the torus $T_{\bar{v}}$ given by the closure ...
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Representations of reductive Lie group
Let $G$ be a reductive algebraic group and $\varrho$ a representation of $G$ in $GL(n)$. Is it true that $\varrho$ is completely reducible? Moreover, how are related the representations of the Lie ...
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Raising and lowering operators for SL(n,K) on homogenous polynomials
The short version:
Can the theory of weights for SL(n,C) be explained concretely in terms of raising and lowering operators on spaces of polynomials?
A deleted question asked how to prove SL(3,C) ...
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Reductive Lie algebra of a Lie group
In the answer of my question:
On the full reducibility of representations of reductive Lie algebras
James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in ...
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Which compact groups have finitely many irreducible representations of each dimension?
If my understanding is correct, this is true of sufficiently nice nonabelian Lie groups (see Ben Webster's answer below), and any finite group. On the other hand, this is false for any infinite ...
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Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology
The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let'...
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Parallel translation in Lie groups
Let G be a Lie group with a left invariant metric. If X and Y are left invariant vector fields and [X,Y]=0, then it is easy to show that Y is parallel to exp(tX).
But if [X,Y] is not zero, what is ...
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How Can I Tell when A Subgroup of a Lie Group is Generated by Unipotents?
I'm trying to understand the proof of the Oppenheim conjecture using Ratner's theorem, and I don't immediately see why $SO(2,1)$ is generated by unipotents. Why is $SO(2,1)$ generated by unipotents? ...
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Relation of Lie Groups and Cohmology Theories via Formal Group Laws
There is a standard process (for example explained here) to obtain a formal group law form a complex oriented cohomology theory.
For a Lie group G one can choose coordinates at the unit and expand ...
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Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent?
I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group.
Is it true? Where can I find a proof?
A counterexample for open subsets of $...
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For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?
The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case ...
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Representations of \pi_1, G-bundles, Classifying Spaces
This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first ...
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Question on transversal slice of Lie group
Assume we have action of Lie group $G$ on a manifold $X$. Fix some orbit $\mathcal{O}$, it is known there exist transversal slice $S$ with respect to this orbit. Fix some point $x$ in $\mathcal{O}$, ...
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The space of Lie group homomorphisms
Let $\ \mathrm{Hom}(H,G)\ $ be the space of Lie group homomorphisms between compact connected Lie groups $H$, $G$. What is known about homology (or homotopy) groups of $\mathrm{Hom}(H,G)$?
UPDATE: $...
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Fourier transforms via Kurzweil-Henstock integral on locally compact commutative groups
Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?
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Invariant Vector Fields for Homogenous Spaces
As we all know, the space of invariant vector fields on a Lie group can be identified with the tangent space at the identity (or any other point for that matter). My question is: How does this ...
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Occurrence of the trivial representation in restrictions of Lie group representations
Suppose $G$ is a semisimple group, and $V_{\lambda}$ is an irreducible finite-dimensional representation of highest weight $\lambda$. Suppose $H \subset G$ is a semisimple subgroup. What is the ...
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