# 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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### Gram-Schmidt map as a Riemannian submersion

We equip $\mathrm{GL}(n,\mathbb{R})$ and $\mathrm{O}(n)$ with their left-invariant metrics, whose restrictions to the corresponding neutral elements is the standard inner product $\mathrm{Trace}(AB^{\...

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### Real points of reductive groups and connected components

Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...

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### An easier reference than “On the Functional Equations Satisfied by Eisenstein Series”?

I'd like to learn about Eisenstein series so I started reading "On the Functional Equations Satisfied by Eisenstein Series"by Langlands.
http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/...

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### Basic notation question involving Lie Groups and Lie algebras

I just started reading "On the functional equations satisfied by Eisentstein series" by Langlands http://publications.ias.edu/sites/default/files/Eisenstein-ps.pdf . I wasn't sure of some notation/...

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### Compact image of adjoint action

Let $G$ be a connected Lie group and $H$ a connected Lie-subgroup.
Suppose that for every compact subset $K\subset G/H$ the $H$-orbit $H.K$ is relatively compact in $G/H$.
Is it true that the image ...

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### $SO(m+1)$-equivariant maps from $S^m$ to $S^m$

Let $G=SO(m+1)$ , $m \geq 2$, act in the standard way on $S^m$.
Let $F:S^m \to S^m$ be a $G$-equivariant map, i.e., $g F(g^{-1}x) =F(x)$ for all $x \in S^m$ and $g \in G$.
Question 1: Is F the ...

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### Simple modules for direct sum of simple Lie algebras

I think that the following statement is true, but I do not know how to prove it.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...

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### Homotopy classes of maps between special unitary Lie group. Correction [duplicate]

An hour ago I asked a question (under the same title) but I used a wrong notation. Here is the improved version.
We consider a special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and ...

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### Homotopy classes of maps between special unitary Lie groups

I am sorry to mislead the notations: $SU(n)$ should be replaced by $PSU(n)$. I will reformulate it now.
We consider the special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and we ...

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### Homotopy of group actions

Let $G$ be a topological group and $X$ be a topological space.
Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous ...

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### More general form of Fourier inversion formula

My question begins as follows: Suppose $G$ is a compact (Lie) group and $A$ is a $G$-module, and denote the action of $G$ on $A$ by $\alpha$. Fix $a\in A$ and view
$$
f:g\mapsto \alpha_g(a)
$$
as an $...

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### Definition of a Dirac operator

So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as:
$D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...

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### A converse of Cartan's automatic continuity theorem

Let $G$ be a compact real Lie group. We say that $G$ has property $(*)$ if every abstract automorphism of $G$ is continuous. A theorem of Cartan says that if $G$ has perfect Lie algebra, it has ...

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### Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions

Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:
$$
1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1
$$
There exists a ...

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### Extensions of compact Lie groups

Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups
$$
0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0,
$$
$$
0\rightarrow G\...

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### Compact connected Lie groups isomorphic as groups and manifolds

Let $G_1$ and $G_2$ be compact connected (not necessarily semi-simple) Lie groups. Assume that the underlying smooth manifolds of $G_1$ and $G_2$ are diffeomorphic and that the underlying abstract ...

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### Branching to Levi subgroups in SAGE and the circle action

In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/...

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### GAP versus SageMath for branching to Lie subgroups

Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
...

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285 views

### Geometric (smooth) Rubik's cube

I and my friend are thinking about a smooth analog of Rubik's cube. One idea is the following:
Consider the 2-dimensional sphere $S^{2}$. We choose three parameters: $(L, H, \theta)$. Here $L$ is a ...

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### Lagrangian subgroup of a nonabelian Lie group

My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory.
See a previous post for other background ...

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### Can a hyperbolic manifold be a product?

I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...

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### Decomposition of the group of Bogoliubov transformations

Consider the fermion Fock space $\mathcal{F}=\bigoplus_{k\ge 0}\bigwedge^k\mathfrak{h}$ of some finite-dimensional 1-particle Hilbert space $\mathfrak{h}$. The group $\mathrm{Bog}(\mathcal{F})$ of ...

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### Does $SU(N)$ have pseudo-real representation?

For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real irreducible representation? (The adjoint representation of $SU(N)$ is real).
A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\...

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### Termination of Cartan's equivalence method

The Cartan-Kuranishi theorem guarantees that a PDE or EDS can always be completed, by prolongation, to involution. My question is, and this is quite murky to gauge from the literature, whether Cartan'...

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### Possible compatible “”structure" on the Lie algebra associated with a Lie group with respect to group structure

For simplicity, let us work on an example.
Regard $GL_r(\mathbb C)$ as a Lie group with associated Lie algebra $M_r(\mathbb C)$, then there exists a canonical so called exponential map:
$$\exp: M_r(\...

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### Is every Lie subgroup of a Lie group isometric to all its conjugates?

Let $G$ be a Lie group with a left invariant metric. Assume that $N$ is a Lie subgroup of $G$.
For a given $g\in G$, are $N$ and $g^{-1} N g$ necessarily isometric Riemannian manifold when they ...

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### Associated bundle construction and classifying space

Let $\theta:G\rightarrow H$ be a morphism of Lie groups.
Given $G$ we have classifying space $BG$ and given $H$ we have classifying space $BH$. This $\theta:G\rightarrow H$ gives a map $B\theta:BG\...

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### Condition on a Lie groupoid to be represented by manifold/group or an action groupoid

Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions.
When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some ...

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### Homeomorphism type of the horofunction boundary for nilpotent Lie groups

Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form
$$\beta_y(x)=d(x,y)-d(w,y).$$
The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...

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### Metrically homogeneous spaces as inverse limits

Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following:
Is $X$ ...

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### Heisenberg groups, Carnot groups and contact forms

The horizontal distribution in the Heisenberg group is the kernel of the standard contact form:
$$
\alpha = dt + 2 \sum_{j=1}^n (x_j \, dy_j - y_j \, dx_j).
$$
Question. Can one describe ...

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### Reference Request: Structure constants for G2

Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...

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### generalization of the discreteness of Hecke groups to general reductive groups

Consider the subgroup $G_{\lambda}$ of $SL_2(\mathbb R)$ generated by $N_{\lambda} = \begin{bmatrix} 1 & \lambda \\ 0 & 1 \end{bmatrix}$ and $S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{...

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### Average of product of matrix elements in irreducible representations of unitary groups

Let $\mathcal{U}(N)$ be the unitary group.
It is well known that
$$ \int_{\mathcal{U}(N)} U_{ij} U^\dagger_{nm} \,dU=\delta_{im}\delta_{jn}\frac{1}{N},$$
where $dU$ is the Haar measure.
More ...

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318 views

### Homogeneous Riemann Surfaces

A Riemann surface $X$ is a connected complex manifold of complex dimension one. A homogeneous space is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification ...

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### On uniqueness of mapping preserving triality

I'm reading Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society 39 (2): 145–205 regarding $\mathrm{Spin}(8)$ and trialities. My question is about this sentence:
The ...

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### Equivalence of Lie theoretic definitions of an arithmetic lattice

In Margulis' book, there are actually two definitions of arithmetic lattices:
If $\mathbf{G}$ is a connected semisimple algebraic $\mathbb{R}$-group, then a lattice $\Gamma \subset \mathbf{G}(\mathbb{...

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### Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?

Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...

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### Does the pushforward of the Haar measure of a semisimple compact Lie group along a character determine the character?

Let $G$ be a connected compact semisimple Lie group.
Let $V$ be a faithful representation of $G$,
with character $\chi \colon G \to \mathbb{C}$.
Let $\mu_G$ be the normalized left Haar measure. (So $\...

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### Better names for Lie groups

After reading this question I was wondering whether mathematicians tried to invent better names for exceptional simple Lie groups $F_4, E_6, E_7, E_8$ ? These names seems a bit obscure and does not ...

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### Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight

Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight.
It is an exercise of Bröcker's book on Representations of Compact Lie ...

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### self-dual integral transform

Is it possible to describe all integral transforms where the inverse transform is implemented by the same formula (with maybe a sign flipped somewhere). Fourier is such an example obviously. I am ...

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### Inductive limits of unitary groups and quantum mechanics

I'm curious if someone has seen concrete applications of $U(\infty)$ in quantum mechanics. Is it possible, for example, in some particular cases to write down the propagator as a limit of a sequence ...

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### How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?

Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...

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### Embedding of Riemannian symmetric spaces $E_I$ and $E_{IV}$ into Lie group $E_6$

In answer and comments to this mathoverflow question we have discussed possiblity of embedding Riemmanian symmetric spaces $E_I, E_{II}, E_{III},E_{IV}$ of dimension $42,40,32,26$ respectively into $...

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### Orbits of unipotent groups

Let $V$ be a real vector space (of finite dimension) and let $G$ be a unipotent Lie subgroup of $\mathrm{GL}(V)$. The orbits of points under the action of $G$ (that is, the sets $Gx = \{T(x) \ : \ T \...

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### Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\rightarrow BH$?

Given a morphism of Lie groups $ \theta:G\rightarrow H$ and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle.
See that the morphism of Lie ...

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### Flag manifolds as incidence correspondences

Let $G$ be a reductive group, $B$ a Borel and $P_j$ the maximal parabolics, indexed by the vertices $j$ of the Dynkin diagram. Then $B = \bigcap_j P_j$, so the flag manifold $G/B$ injects into $\...

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### Commutator length in connected solvable Lie groups

Let $G$ be a connected solvable Lie group and let $H$ denote ist commutator subgroup. By definition, every element $g \in H$ can be written as a product of commutators and the minimal number of ...

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### Is conjugation by gauge transformation of $G$-bundle contained in $\mathfrak{g}$?

Before painstakingly defining all these terms, let me ask my question in plain english: given a $G$-bundle, is every conjugate of a vector field by a gauge transformation an element of the Lie algebra?...