Questions tagged [lie-groups]

Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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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 ...
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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 \...
Mini's user avatar
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2 votes
1 answer
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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 $G$,...
unknownymous's user avatar
7 votes
1 answer
316 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 ...
Dominic Else's user avatar
1 vote
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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 ...
ABIM's user avatar
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6 votes
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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 ...
Ali Taghavi's user avatar
4 votes
1 answer
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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: ...
sleeve chen's user avatar
-1 votes
1 answer
98 views

Topological connected eccentrics, not homeomorphic to commutative Lie groups

An eccentric is a universal algebra $\ (X\ \sigma\ \lambda\ \rho)\ $ such that operations $\ \sigma\ \lambda\ \rho\,:\,X\times X\to X\ $ satisfy: $\quad \forall_{x\ y\,\in X}\quad \lambda(\sigma(x\ y)...
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When a free action gives rise to a $G$-principal bundle

When a free action gives rise to a $G$-principal bundle Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that $G \backslash X$ is Hausdorff. (equivalently the image of ...
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Confusion over spin representation and coordinate ring of orthogonal Grassmannian

This is a copy from MSE where the question did not attract much attention. I'm working over $\mathbb{C}$ here. Let $G=\mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the maximal parabolic ...
Sam Hopkins's user avatar
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Small pants in arithmetic hyperbolic surfaces of high degree

Does the following statement hold: Statement: For any $\epsilon > 0$, there exist a number field $k$ of degree $d_{\epsilon}$ over $\mathbb{Q}$ and an arithmetic hyperbolic surface  $\Gamma$ ...
shurtados's user avatar
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8 votes
2 answers
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Ideals generated by regular sequences

In Vasconcelos' paper (Ideals generated by R-sequences), he proved If $R$ is a local ring, $I$ an ideal of finite projective dimension, and $I/I^2$ is a free $R/I$ module, then $I$ can be ...
Cubic Bear's user avatar
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Totally geodesic submanifolds of SO(3) [closed]

Consider the special orthogonal group $SO(3)$ with its bi-invariant metric (or equivalently, with the metric induced by its standard embedding to the space of $3\times 3$ real matrices). Obviously, $...
C M's user avatar
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Bound between distance between Rotation Matrices

Let $\|\cdot\|_F$ denote the Fröbenius norm on the set of $d\times d$ matrices. By restriction this induces a metric on $SO(n)$. Let's make an observation. Since $X\in SO(n)$ is a rotation matrix ...
ABIM's user avatar
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A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$

This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there or here. $G$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie ...
Chertopkhanov on Malek Adel's user avatar
5 votes
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Reference request: Name or use of this group of diffeomorphisms of the disc

Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following: $ \phi(S_r^...
ABIM's user avatar
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3 votes
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Convergence of Fuchsian groups and existence of suitable homeomorphisms

Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...
Jongar Jongar's user avatar
1 vote
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Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms

Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map $$ ...
ABIM's user avatar
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3 votes
1 answer
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Continuity/Lipschitz regularity of exponential map from $C_c$ to $\operatorname{Diff}_c$?

For finite-dimensional Lie algebras, see this for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to ...
ABIM's user avatar
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5 votes
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What is the name of the real form corresponding to the quaternionic symmetric space?

Let $G$ be a compact simple Lie group. Choose a system of positive roots, and let $\mathrm{SU}(2) \subset G$ correspond to the highest root, and $\mathbb{Z}/2 \subset \mathrm{SU}(2)$ the centre. The ...
Theo Johnson-Freyd's user avatar
11 votes
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The existence of a fiber sequence involving $\mathrm{Spin}(9)$ and $\mathrm{SU}(2)$

$\newcommand{\SU}{\mathrm{SU}} \newcommand{\Spin}{\mathrm{Spin}}$There is a fiber sequence $G_2\to \Spin(9) \to \Spin(9)/G_2$, and $G_2$ contains $\SU(3)$ as a subgroup. Is there a space (possibly a ...
skd's user avatar
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28 votes
2 answers
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What is special to dimension 8?

Dimension $8$ seems special, as the partial list below might indicate. Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$? Surely it isn't it, in the end, simply because ...
1 vote
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466 views

Horizontal lift of fundamental vector field

Suppose $\theta\colon G\times M\to M$ is a transitive smooth left action of a compact Lie group $G$ on a manifold $M$ and $\pi\colon G\to M\cong G/K$ the corresponding smooth submersion for some ...
Sven Pistre's user avatar
2 votes
0 answers
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Realization of limit of discrete series using Dirac operators

I wonder if there is a geometric realization of limit of discrete series in the flavor of Atiyah-Schmid or Parthasarathy realizing discrete series using Dirac operators on G/K. I know you can see ...
GTA's user avatar
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3 votes
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Existence of a maximal rank CR Lie subalgebra

Let $\mathfrak{g}$ be a real Lie algebra of dimension $2n+1$ and let $\mathfrak h \subset \mathfrak g \otimes \mathbb C$ be a subalgebra of complex dimension $n+1$ satisfying $\mathfrak h + \overline{\...
Max Reinhold Jahnke's user avatar
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Isoclinism for Lie groups: existing accounts of basic properties?

Philip Hall introduced the relation of isoclinism between two groups. One statement of the definition (not Hall's original statement) is to introduce a category whose objects are the canonical maps $$...
Yemon Choi's user avatar
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Is there an orbit map without path lifting property?

I am looking for an example of a topological group $G$ acting by homeomorphisms on a metrizable space $X$ such that the orbit map $X\to X/G$ doesn't have the path lifting property, that is, there is a ...
Igor Belegradek's user avatar
7 votes
2 answers
392 views

Homomorphisms from higher rank lattices with infinite center to $\mathbb{Z}$

Suppose that $\Gamma$ is an irreducible lattice in a semi-simple real Lie group $G$ of higher rank (with infinite center!), is every homomorphism $\Gamma \to \mathbb{Z}$ trivial? The case where $G$ ...
shurtados's user avatar
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2 votes
2 answers
393 views

Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$

Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
ABIM's user avatar
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3 votes
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Diffeomorphisms fixing origin and boundary

Let $D^n$ be a disc in $\mathbb{R}^n$. Is there a known characterization of all the diffeomorphisms of $D^n$ fixing the origin and boundary of $D^n$?
ABIM's user avatar
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Kazhdan Property T of semisimple Lie groups

I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M., Analogs of Wiener's ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), no. 2, 233–259] (MSN). I want to ...
A beginner mathmatician's user avatar
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123 views

Diffeomorphisms of a "matrix type"

Let $\exp$ denote the matrix exponential map, let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ be defined the collection of all functions of the form $$ f(x) = \exp\left( \sum_{i=1}^n f_i(x) A_i \...
ABIM's user avatar
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-1 votes
1 answer
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A manifold or Riemannian structure on the space of all conjugacy classes of a compact Lie group [closed]

Let $G$ be a compact Lie group. Is each conjugacy class a closed subset of $G$? Define the conjugacy equivalent relation $g\sim h$ if $g$ is conjugate to $h$.Is $G/\sim$ a Haussdoef space with ...
Ali Taghavi's user avatar
3 votes
0 answers
56 views

Criteria for density of subgroup of diffeomorphism group

Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
ABIM's user avatar
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2 votes
1 answer
125 views

Lifting one parameter subgroup $e^{t K}$ to the universal cover of $\mathrm{Sp}(2N,\mathbb{R})$

I would like to lift an arbitrary one-parameter subgroup $e^{t K}$ with $K\in\mathfrak{sp}(2N,\mathbb{R})$ to the universal cover $\widetilde{\mathrm{Sp}}(2N,\mathbb{R})$ (or at least its two-fold ...
LFH's user avatar
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2 votes
1 answer
391 views

Dense subgroup of a compact connected Lie group generated by two words

Let $G$ be a compact connected Lie group and $w_1$, $w_2$ be two positive words in alphabet $\{a, b\}$ which are not the powers of some another word $w$. Positive means that $a^{-1}$ and $b^{-1}$ can ...
Dmitri Scheglov's user avatar
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Gradient descent in $U(n)^r$

I have a function $f:U(n)^r\rightarrow \mathbb{R}$ which I would like to minimize. Here, $U(n)$ is the set of unitary matrices, and $r$ should be considered to be much bigger than $n$. For instance, $...
Springberg's user avatar
6 votes
1 answer
451 views

Is the connected centralizer of a semisimple element in a connected reductive group also a centralizer?

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field and let $g\in G$ be semisimple. Write $C=\mathrm{C}_G(g)$ and $C^\circ=\mathrm{C}_G(g)^\circ$ for the ...
kneidell's user avatar
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1 vote
0 answers
429 views

When two isomorphic subgroups are conjugate?

I would like where could I find a reference in which this question is answered. Let us consider $H,H'\leq G$ two isomorphic subgroups of a Lie group $G\in \{\mathbf{SO}(n),\mathbf{SU}(n),\mathbf{Sp}(...
J. Salieri's user avatar
2 votes
3 answers
278 views

Space of representations of surface group into Lie groups

In the context of Goldman's paper The symplectic nature of fundamental groups of surfaces: Consider a closed oriented surface $S$ with fundamental group $\pi$, and let $G$ be a connected Lie group. ...
Zineb mazouzi's user avatar
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0 answers
112 views

Commensurability of arithmetic, irreducible, nonuniform lattices

Let $n \in \mathbb{Z}_{\geq 2}$ be arbitrary. Let $r_1$ and $r_2$ be arbitrary elements of $\mathbb{Z}_{\geq 0}$ that satisfy $r_1 + r_2 > 0.$ Let $G := {\rm SL}_n(\mathbb{R})^{r_1} \times {\rm SL}...
Mishel Skenderi's user avatar
3 votes
1 answer
149 views

Morphisms of two fully reducible representations of a group [closed]

I thought this is a simple question and placed it at math.stackexchange.com: https://math.stackexchange.com/questions/3616661/morphisms-of-two-fully-reducible-representations-of-a-group. Since no one ...
Michael_1812's user avatar
2 votes
1 answer
190 views

Orbital integral in Cluckers and Denef

My questions concerns Definition 1.2 of an orbital integral in the paper Orbital integrals on General Linear Groups by Cluckers and Denef. I will recall the definition below, but my question is: how ...
Tian An's user avatar
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3 votes
0 answers
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Choosing a complementary Lie subalgebra well

I have a compact connected Lie group $G$ with Lie algebra $\mathfrak{g}$ and a codimension one closed subgroup $H$ with Lie algebra $\mathfrak{h}$. Using an inner product on $\mathfrak{g}$ one can ...
R Mary's user avatar
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7 votes
0 answers
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What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?

Here is the story as I see it. Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
Tim Phalange's user avatar
4 votes
4 answers
457 views

Algebra of regular functions on the quadratic cone and SU(2) representations

I was reading the paper "Short Star-Products for Filtered Quantizations" by Pavel Etingof and Douglas Stryker (MSN), where in the introduction they claim that the algebra of regular functions on the ...
Simone Castellan's user avatar
4 votes
0 answers
122 views

Closed subgroup (Cartan) theorem without transversality nor Lipschitz condition within Banach algebras

Yesterday, I came across the following preliminary theorem. Theorem Let $\mathcal{B}$ be a Banach algebra (with unit $e$) and $G$ be a closed subgroup of $\mathcal{B}^{-1}$ (the group of ...
Duchamp Gérard H. E.'s user avatar
4 votes
0 answers
66 views

The weak restriction of the Jacquet module

Let $P= MN$ be a parabolic subgroup of a reductive p-adic group $G$, and $(\pi, V)$ is an irreducible, admissible representation of $G$. The Jacquet module is the representation $(\pi_N, V_N)$, where $...
IMED's user avatar
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3 votes
3 answers
533 views

Real representations of SO(n) and U(n)

I would like to get some references where I can find the theory of the real representations of $\mathbf{SO}(n)$ and $\mathbf{U}(n)$. In particular, I would like to know for which dimensions there ...
J. Salieri's user avatar
2 votes
0 answers
37 views

Classification of involutions on $G_{2}$-homogeneous spaces

Are you aware of a systematic classification of involutions on $G_{2}$-homogeneous spaces?
Nicolas Boerger's user avatar

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