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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Which maximal closed subgroups of Lie groups are maximal subgroups?
Which maximal closed subgroups of Lie groups are maximal subgroups?
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Integral subgroup theorem for Banach-Lie groups
Let $G$ be a Banach-Lie group with Lie algebra $\mathfrak g$
and $\mathfrak h$ a closed subalgebra. Using the exponential
map and the Baker-Campbell-Hausdorff-formula one constructs a local
Lie group $...
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Cohomology of lattice subgroups
I am trying to find a reference for lower cohomology groups $H^i(G, \mathbb{Z}),$ for $i=1, 2, 3$ for lattices in higher rank (for example, $SL(n, \mathbb{Z}), Sp(2n, \mathbb{Z}),$ and possibly ...
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Which norms have rich isometry groups?
Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of all bijections $F:\...
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Meaning of "Compact" in 1932 Paper by van der Waerden "Continuity Theorem for Semisimple Lie Groups".
I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions.
I am attempting ...
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When is a finite dimensional real or complex Lie Group not a matrix group
I have a smattering of knowledge and disconnected facts about this question, so I would like to clarify the following discussion, and I also seek references and citations supporting this knowledge. ...
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Constructing solutions to matrix equations
Let $k,n$ be integers, $u_1,\dots,u_n \in U(k)$, $d_1,\dots,d_n \in \mathbb Z$ with $\sum_{i=1}^{n} d_i =: d \neq 0$.
Consider the map $w:= U(k) \to U(k)$ with
$$w(v):= u_1 v^{d_1} \cdots u_n v^{d_n} ...
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lowest weight representation of loop groups
I am trying to understand lowest representations of loop groups as developed in Pressley and Segal's book. Specifically I want to be able to compute the weight spaces that appear in a lowest weight ...
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Are there explicit formulas for spherical functions on oriented real grassmannians?
Let $p$ and $q$ be integers. The group $K=SO(p) \times SO(q)$ can be naturally seen as a subgroup of $G=SO(p+q)$. The quotient space $G/K$ is identified with the space of oriented $p$-dimensional ...
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Is every Lie subgroup of GL(V) isomorphic to a (maybe another) closed subgroup of GL(V)?
I am gathering material for an exposition and I note that some texts (e.g. Ise and Takeuchi, "Lie Groups I & II", Stillwell, "Naive Lie Theory", Hall, "Lie Groups, Lie Algebras, and ...
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What groups are Lie groups?
We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...
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Euler class of S^1-orbibundle
Given a topological space $M$ with a locally free $S^1$ action on it, assume the slice representation holds,(this is often the case, e.g. M is a smooth manifold) then this will make $M$ a principal $S^...
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What would you want on a Lie theory cheat poster?
For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...
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maximal tori cover compact Lie group
Let $G$ be a compact connected Lie group, $T$ be some maximal torus in $G$ (that is, inclusion-maximal connected abelian subgroup). Then the union of tori $gTg^{-1}$, $g\in G$, is the whole $G$. This ...
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Lie group operation and tangent vectors
Suppose we have two differentiable paths $\alpha$ and $\beta$ thru the identity of a Lie group $G$, $\alpha(0)=\beta(0)=e$ the identity element. Denote $\alpha\beta$ the path given by $\alpha\beta(t)=\...
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Is $PSL(n, Z)$ isomorphic to a subgroup of $GL(n,C)$ or even $GL(n+1,C)$?
Is $PSL(n, \mathbb Z)$ isomorphic to a subgroup of $GL(n,\mathbb C)$ or even $GL(n+1,\mathbb C)$?
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Complex Lie group without faithful real representations?
Does there exist a complex analytic Lie group which doesn't have faithful representations in $GL(N,\mathbb R)$, viewed as a real Lie group?
There are examples of complex Lie groups which do not allow ...
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In a compact lie group, can two closed connected subgroups generate a non-closed subgroup?
Let $H$,$K$ be closed connected subgroups of a compact Lie group $G$. Let $L:=\langle H,K \rangle$ be the subgroup they generate, ie, the smallest subgroup of $G$ containing them both. Must $L$ be ...
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Can a Lie group as an abstract group be given more than one topology making it a Lie group?
I am an optical engineer, so please forgive any ignorance my questions betoken. I am interested in whether one can tear down the manifold of a finite dimensional Lie group,
leaving an abstract group, ...
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On closed totally disconnected subgroups of connected real Lie groups
So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that ...
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cohomology of BG, G compact Lie group
It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group
$G$. However, I've still not found a proof of this. I believe that the proof is as follows:
--> $G$ compact ...
- 1,709
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"locally" factoring subgroups of Lie groups
I'm not really a math person, and apologize if the question here is too simple. I've ended up with the following type of question for a few Lie groups, but state it for SO(n).
I start with a subgroup ...
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answers
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weyl group representations
I am looking for some references about irreducible representations of the Weyl Group over simple Lie Groups, both classical and exceptional ones. In particular I want to know the dimensions and the ...
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Generator of Lie Group cohomology in degree 3
This is my first question.
Take a simple, connected, compact, simply connected Lie group $ G$ (dim $G\geq 3$).
The cohomology of $G$ with integer coefficients is
$H^{1,2}(G,\mathbb{Z})\cong 0$, $H^{...
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answers
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Fixed points of the action of an algebraic group
Hello!
If a compact Lie group $K$ acts smoothly on a smooth manifold $M$, then the set $M^K$ of fixed points under this action is a smooth submanifold of $M$. This is proved for example in ...
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There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?
How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) \...
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References on Lie groups and dynamical systems
I'm interested in Lie theory and its connections to dynamical systems theory. I am starting my studies and would like references to articles on the subject.
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Modern reference for maximal connected subgroups of compact Lie groups
What's the nicest place to see a list of the maximal connected subgroups of compact Lie groups? Is there anything on-line?
I looked at Tits' Bourbaki talk on Dynkin's and others' work, but he admits ...
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Explanation of $y = x \exp(\triangle)$ for a Lie Group
Let $M$ be a non-compact matrix Lie group and $T_e M$ its lie algebra.
Consider a point $x \in M $ and $ \triangle \in T_e M$.
To move from $x$ to a point $y \in M$ along $\triangle$, below group ...
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Orbits of semi-algebraic actions
Hello all,
I recently came across the following Theorem in Gibson (Singular points of smooth mappings, 1979). Since I haven't seen this result somewhere else and this reference is not so widespread, ...
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467
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Difference between action of group element and Lie algebra element in smooth representation
Let $G$ be a real reductive Lie group, $P$ its parabolic subgroup with Levi decomposition $P=MN$, let $\mathfrak{n}$ be the nilpotent Lie algebra of $N$. Suppose given a smooth representation $(\pi,V)$...
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Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$
The equivalence I describe below is well-known, but I'd like a simple standard reference for it.
Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
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509
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Small sum of group elements acting as rank 1 matrix.
I am interested in constructing small (as possible) group $G$ with large dimensional irreducible representation $\rho,V$ such that exist three elements of $g_1,g_2,g_3\in G$ such that for some $c_1,...
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reference containing the list of irreducible finite dimensional representation of real general linear group
It seems that it is not easy to find a reference containing a classification and construction of finite dimensional irreducible representations of $GL_n(\mathbb{R})$. One way to look at it is via $(\...
- 2,709
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641
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Looking for general approaches to show connectedness of topological groups
Let $G$ be a topological group. One general approach to show that $G$ is connected is the following:
For every subgroup $H\leq G$ (not necessarily closed) we have a projection map:
$$
\pi: G\...
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Intersections of conjugates of the icosahedral group in SO(3)
(Related question)
Let $I$ be the group of orientation preserving symmetries of a regular icosahedron. This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternating group $A_5$. It is ...
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nullity of the second fundamental group of a Lie group
Can anyone tell me why it is that Lie groups seem to have their second fundamental group $\pi_2(G)$ equal to $0$, or provide me with a link to an article or a book reference?
I came across this fact ...
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Why study Lie algebras?
I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
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Does every irreducible representation of a compact group occur in tensor products of a faithful representation and its dual?
(Previously posted on math.SE with no answers.)
Let $G$ be a compact Lie group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. Is it true that every (complex, ...
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Origin of terms "flag", "flag manifold", "flag variety"?
These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably ...
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Intersections of conjugates of Lie subgroups
Let $G$ be a closed, connected Lie group, and let $H$ be a closed (and therefore Lie) subgroup. There is a natural action of $G$ on the space of left cosets $G/H$, for which the stabiliser of $aH$ is ...
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Polynomial group Laws on $\mathbb{R}^2$
When students are first learning about groups, a classic example of a group that is not defined as a set of functions is the group whose underlying set is $\mathbb{R}\setminus-1$, and whose operation ...
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Orbits for homogenous complex polynomials under unitary rotation of variables
Let's have two complex homogeneous polynomials of degree $k$: $f(z_1,\cdots,z_n)$ and $g(z_1,\cdots,z_n)$. We consider rotations of variables in the form of $\vec{z}' = U \vec{z}$, where $U\in SU(n)$.
...
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average Riemannian distance between Identiity and a random point in SO(n) or SU(n)
I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...
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Does $SO(32) \sim_T E_8 \times E_8$ relate to some group theoretical fact?
It is well known the existence of a T duality between the two heterotic string theories, $SO(32) \sim_T E_8 \times E_8$. Beyond the trivial point that both groups have the same dimension (496, which ...
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Does a symplectic group act on a tensor power of a spin representation?
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$More specifically, let $S_k$ be the spin representation of $\Spin(2k+1)$.
Then is there are action of $\Sp(2r-2)$ on $\bigotimes^{2r}S_k$ ...
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Does Branching in the Weight Diagram affect an embedding?
All groups here are compact semisimple Lie groups. Out of laziness I will use $B_7$ to mean $Spin(15)$.
Suppose that one has a group $H$ and a subgroup $G$. The embedding determines the decomposition ...
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What is the group O(4)/H, where H is the center of O(4)?
What is the group $O(4)/H$?
Here $O(4)$ is the group of orthogonal matrices and H is the center of $O(4)$.
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What does a homogeneous space of a linear algebraic group know about the group?
Let $X=G/H$, where $G$ is a connected linear algebraic group over the field $\mathbf{C}$ of complex numbers
and $H\subset G$ is an algebraic subgroup.
In general, we can write the algebraic variety $X$...
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different Shimura data with common underlying group?
A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge ...
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