Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
2,189
questions
6
votes
1answer
68 views
Gelfand pair, weakly symmetric pair, and spherical pair
I am a bit confused with the relations among Gelfand pairs, weakly symmetric pairs, and spherical pairs defined in the book "Harmonic analysis on commutative spaces" written by professor ...
2
votes
1answer
86 views
Lie monoids as monoids internal to the category of smooth manifolds?
This question can be thought as a complement to this one.
Lie groups can be defined as groups internal to the category of smooth manifolds. Lie monoids, however, as a particular case of Lie semigroups,...
5
votes
0answers
176 views
+50
Interpolated simple integral fusion categories of Lie type
$\DeclareMathOperator\PSL{PSL} \DeclareMathOperator\Rep{Rep}$The idea motivating this post is that there should exist a global understanding of the unitary fusion categories $\Rep(G(q))$, with $G(q)$ ...
2
votes
0answers
93 views
Explicit tensor product decomposition for the representations of PSL(2,q)
$\DeclareMathOperator\PSL{PSL}$Let the type of the character theory of a finite group $G$ be the list $[[d_1,n_1], \dotsc, [d_k,n_k]]$ with $1=d_1 < \dotsb < d_k$ and $n_i$ the number of ...
8
votes
1answer
491 views
Learning from unsuccessful attempts at the Poincaré conjecture
This is reposted from MSE, but perhaps it is more appropriate to post it here. Let me know if I'm wrong.
Among the many unsuccessful attempts at solving the Poincaré conjecture, I'm wondering if there ...
4
votes
0answers
159 views
Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$We know that
$$
\Spin(1,3)=\SL(2,\mathbb C)
$$
and
$$
\Spin(4)=\SU(2) \times \SU(2).
$$
The $\Spin(1,3)$ is the ...
1
vote
1answer
63 views
Eigenvectors of random unitary matrices
Any unitary matrix $U$ can be diagonalized by another unitary matrix $V$,
$$U=VDV^\dagger,$$
where $D={\rm diag}(z_1,z_2,...,z_N)$ is diagonal.
If $U$ is taken at random uniformly with respect to Haar ...
18
votes
2answers
445 views
What is known about the “unitary group” of a rigged Hilbert space?
Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
1
vote
0answers
27 views
Small deformations of maximally symmetric 3-spaces
I am looking for all 'small' deformations of the three 3-dimensional Riemannian spaces with maximal symmetry, the pseudo-sphere, Eucidean space and the sphere. By a theorem by Fubini (1903) the ...
3
votes
0answers
60 views
A pseudo-Riemannian version of a theorem by Fubini
Guido Fubini, ``Sugli spazii che ammettono un gruppo continuo di movimenti,'' Annali di Mat., ser. 3, 8 (1903) 54.: Let $M$ be a Riemannian manifold of dimension $d\ge 3$. Its isometry group cannot be ...
3
votes
0answers
70 views
On the Gelfand-Kirillov Conjecture
The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...
11
votes
3answers
695 views
Smooth map homotopic to Lie group homomorphism
Let $G$ and $H$ be connected Lie groups. A Lie group homomorphism $\rho:G\to H$ is a smooth map of manifolds which is also a group homomorphism.
Question: Can we find a smooth (or real-analytic) map $...
5
votes
1answer
111 views
Equivalence generated by Jacobian minors
Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ ...
1
vote
1answer
43 views
Let $G'\triangleleft G<\operatorname{Iso}(M)$ be a normal subgroup. A $G'$-stratum is the union of $G$-strata of lesser dimension
Let $G$ be a group of isometries acting effectively by isometries on a connected Riemannian manifold. And let $G'\triangleleft G$ be a normal subgroup. I am trying to prove that $\dim \operatorname{St}...
3
votes
0answers
85 views
Let $T$ be a maximal torus of $SU(k+1)$. Who is the normalizer $N(T)$ of $T$ in $O(2k+2)$?
I'm reading an article which I cannot understand a paragraph very well.
$T$ is a maximal torus of $SU(k+1)$ acting linearly on $\mathbb{C}^{k+1}$. And here is what is written that I cannot fully ...
7
votes
2answers
562 views
Élie Cartan's paper “Les groupes réels simples, finis et continus” of 1914
Question 1.
Does Élie Cartan's paper
Les groupes réels simples, finis et continus,
Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355
contain a classification of $\Bbb C$-linear involutions of simple ...
2
votes
0answers
83 views
Jordan normal form in a reductive group
Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
4
votes
1answer
86 views
Regarding extensions of finite groups by Tori
I know how to prove the following result. However, my proof is a little bit long and complicated and only uses fairly low tech results in group cohomology. It would be nice if I could find a citation ...
4
votes
1answer
129 views
Average of product of matrix elements in the special orthogonal group
Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed, in Integration with respect to the Haar measure on unitary, orthogonal and symplectic group (see ...
2
votes
0answers
38 views
Natural appeareances of (commutative) algebras in $\mathfrak g$-modules
$\newcommand{\g}{\mathfrak g}$
Let $\g$ be a Lie algebra, and observe that since $U(\g)$ is a cocommutative Hopf algebra, it makes sense to look for (naturally arising and perhaps commutative?) ...
7
votes
0answers
220 views
Is every space a classifying space?
Despite a pretty thorough look (I think) I can’t find the answer to the following question: Is every (reasonable?) path connected space weakly equivalent to the classifying space of some topological ...
4
votes
2answers
180 views
How to describe the compact real forms of the exceptional Lie groups as matrix groups?
I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe ...
2
votes
2answers
209 views
Relation between two homomorphisms from $SO(3)$ to the Möbius group $PGL(2,\mathbb{C})$ [closed]
Let $f, g$ be two homomorphisms from $\mathrm{SO}(3)$ to $PGL(2,\mathbb{C})$. Does there exist $S \in \mathrm{PGL}(2,\mathbb{C})$ such that for all $X\in \mathrm{SO}(3)$ we have $g(X)=Sf(X)S^{-1}$?
...
2
votes
0answers
47 views
From restricted root space to root space
I am reading Knapp's book "Lie Groups beyond Introduction" https://link.springer.com/book/10.1007/978-1-4757-2453-0. I do not understand the following argument in Page 377 (2nd edition). I ...
1
vote
1answer
59 views
Iwasawa decompostion and simply connected subgroups [closed]
Let $G$ be a semisimple Lie group, i.e. $G$ is connected and Lie algebra of $G$ is semisimple. We know by Iwasawa decomposition, there are connected subgroups $K$, $A$ and $N$ of $G$ such that the ...
5
votes
0answers
67 views
Is the symplectomorphism group of a compact manifold a tame Fréchet Lie subgroup of $\operatorname{Diff}(X)$?
In the famous paper Hamilton, Richard S. The inverse function theorem of Nash and Moser.
Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222, Hamilton introduced the category of tame Fréchet Lie ...
15
votes
0answers
163 views
Lie theoretic meaning to $e^{\text{cycle}} = \text{permutation}$?
It is well known that exponentiating the EGF(exponential generating function) for cycles gives the EGF for permutations: link here. Usually summarized under the catchy slogan ...
15
votes
2answers
1k views
Emergence of the orthogonal group
Do we know what mathematician first considered, and perhaps named, what we call the group $\mathrm O(n)$, or $\mathrm{SO}(n)$, for some $n>3$?
I mean it specifically as group (not Lie algebra) ...
3
votes
0answers
74 views
Books on integration on semisimple Lie groups
Can anyone suggest me some good books where I can find integration theory on semisimple Lie groups (using KAK, KAN and other type of decompositions)?
I have read Knapp's book "Lie groups beyond ...
2
votes
0answers
66 views
Reductive Lie groups and existence of maximal compact subgroup
I am reading Knapp's book "Lie groups beyond an introduction" (2nd edition). I am struggling to understand the following point. Recall that $G$ is a reductive Lie group. If the Lie algebra $\...
2
votes
1answer
67 views
Explicit Normalizer of SU(3) Cartan subalgebra
The normalizer $N(\mathfrak{h})$ of the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{su}(3)$ is defined as
$$N=\left\{ x \in SU(3)\;|\; x^\dagger\mathfrak{h}x \in \mathfrak{h}\right\}$$
I would like ...
6
votes
2answers
145 views
Universal property of induced representation
Let $H$ be a closed subgroup of the compact Lie group $G$. Let $E$ be a continuous representation of $H$. In the book "Representations of compact Lie groups" by Bröcker and Dieck the induced ...
3
votes
1answer
232 views
Generalization of Killing form
I am reading Knapp's book "Lie groups beyond introduction". On page 369, he has described the following. Let $\mathfrak g$ be a real semisimple Lie algebra. Suppose $\theta\colon\mathfrak g\...
3
votes
0answers
95 views
Moment map of $\mathrm{O}(n)$-action on $\mathbb{C}^n$
Let $(\mathbb{C}^n, \omega_0)$ be the complex Euclidean space of dimension $n$ with the standard Kähler structure $\omega_0$. I am looking for a Hamiltonian $\mathrm{O}(n)$-action on $(\mathbb{C}^n, \...
1
vote
2answers
176 views
On the smoothness of transition functions
Let $p:E \longrightarrow M$ be a smooth fibre bundle, with standard fibre space $F$ and $G$ a Lie group acting effectively on $F$ as a structure group.
Then, are the transition functions always ...
3
votes
0answers
82 views
Decomposing a compact connected Lie group
I want to prove the following. Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ and center $Z_G.$ It is not hard to prove that $\mathfrak g$ is reductive. Therefore, we can ...
4
votes
0answers
115 views
Involutive automorphism of simple Lie algebra
I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange.
Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan ...
3
votes
0answers
80 views
Typical preimage of the commutator map
By Goto's theorem for any compact connected semisimple Lie group $G$ of dimension $n$, any element $x\in G$ is a commutator, namely $x=[y,z]$ for some $y, z\in G$. Another way to say it is that the ...
1
vote
1answer
99 views
Irreducible non-Abelian subgroup of $\mathrm{U}_n(\mathbb{C})$, containing diagonal matrices
Consider an irreducible non-Abelian subgroup $\mathrm{H}$ of group of unitary matrices $\mathrm{U}_n(\mathbb{C})$, that contains the subgroup of diagonal matrices. Does there exist any result ...
4
votes
1answer
156 views
The normalizer of block diagonal matrices
Let $G=\mathrm U_n$ or $\mathrm{GL}_n(\mathbf C)$ and $H$ the subgroup of block diagonal matrices respecting a partition $n=n_1+\dots+n_k$. Is the normalizer $N=N_G(H)$ computed anywhere in the ...
3
votes
0answers
46 views
Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries
Could anyone please suggest related papers or article about the topic related to my one question below?
Reduce PDE to ODE by dilation symmetry
I also cite a paper in the link above.
We know that ...
8
votes
0answers
115 views
Comparison of two well-known bases of the integral homology group of based loop group
Let $G$ be a compact simply-connected Lie group. Then one can look at the homology $H_*(\Omega G;\mathbb{Z})$ of the based-loop space $\Omega G$ in at least two different ways:
(1) Via Bott-Samelson'...
3
votes
0answers
53 views
Conjugacy classes in reductive group under adjoint action of parabolic subgroup
Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results ...
0
votes
0answers
76 views
Can we extend a function from the diagonal matrices to an orthogonally-invariant function on $\text{GL}_n$?
This is a cross-post.
Let $g:(0,\infty)^n \to [0,\infty)$ be a symmetric function -i.e. $g(\sigma_1,\dots,\sigma_n)$ does not depend on the order of the $\sigma_i$, with $g(1,\dots,1)=0$.
We ...
4
votes
1answer
250 views
Smallest subgroup of unitary group, containing diagonal matrices and a fixed unitary matrix is the whole group
Suppose that $U_n(\mathbb{C})$ is the group of unitary matrices of dimension $n$ over complex numbers. Fix a unitary matrix $A \in U_n(\mathbb{C})$ and consider the smallest closed subgroup $K \...
2
votes
1answer
183 views
Conjugacy of elements in a parabolic subgroup
Let $G$ be a complex connected reductive group, and let $P \subseteq G$ be a parabolic subgroup. My question is the following: if $g$ and $h$ are elements of $P$ which are conjugate as elements of $G$,...
6
votes
1answer
131 views
Does a compact Lie group have finitely many conjugacy classes of maximal Abelian Lie subgroups?
Let $G$ be a compact Lie group. An Abelian Lie subgroup $A \leq G$ is a maximal Abelian Lie subgroup if, for any Abelian Lie subgroup $A'$ such that $A \leq A' \leq G$, then $A' = A$.
Of course any ...
5
votes
0answers
37 views
Continuous surjection from $X(D_n)$ onto $\operatorname{Homeo}_0(D_n)$
Let $n>1$ and let $\mathfrak{X}(D_n)$ denote the set of continuous vector fields on the closed disc $D_n\subseteq \mathbb{R}^n$. Let $\operatorname{Homeo}_0(D_n)$ be the set of homeomorphism of ...
6
votes
1answer
221 views
An extension of symplectomorphism group
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$.
We consider the ...
4
votes
1answer
124 views
Reduce PDE to ODE by dilation symmetry
I am reading Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems, p.753.
Consider the following PDE: ...
