# 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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### A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori

$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...

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### Finite subgroups of $\mathrm{SL}_2(\mathbb{O})$

What are the finite subgroups (subloops that are groups) of $\mathrm{SL}_2(\mathbb{O})$? In particular, are there any not contained in a $\mathrm{SL}_2(\mathbb{H})$?

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### Is a Lie subgroup whose center is closed, a closed subgroup itself?

I want to show that a certain Lie subgroup (i.e. generated by the exponential of elements in some Lie subalgebra) of a Lie group is closed. My knowledge of the subject of Lie groups is rudimentary, ...

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### Is there a formula for $fgf^{-1}$ in $G_2$?

Let $G_2$ be the exceptional Lie group of smallest dimension, represented as the exponentials of the antisymmetric real $7×7$ matrices $A$ with $$\sum_{i=1}^{3}{A(x+2^i,x-2^{i-1})} = 0$$ for integer $...

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### Sum of weights of an irreducible representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries.
Firstly, I would like to know ...

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### The largest abelian subgroups of a Lie group [duplicate]

Let $G$ be a semisimple Lie group. Denote by $d(G)$ the maximal integer $p$ such
that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$, and let $c(G)$ denote the maximal integer $q$ such ...

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### Classification of Lie group structures on $\mathbb{R}^n$

Is it possible to describe, up to isomorphism, all Lie groups $G$ whose underlying manifold is diffeomorphic to $\mathbb{R}^n$ (with its standard smooth structure)?
In fact, I haven't found any such ...

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### Complete representation theory of $\mathrm{SL}(2,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...

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### Unitary dual of universal cover

The universal covering group $G$ of $\mathrm{SL}_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\...

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### The largest abelian subgroups of a Lie group

Let $G$ be a semisimple Lie group. Denote $d(G)$ as the maximal integer $p$ such
that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\...

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### Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation
$$
G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) .
$$
Basic ...

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### Why is $\Pi_r^L$ a non-degenerate Poisson structure on $G\ $?

Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...

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### The homotopy type of the space of symplectic structures

While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...

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### Left translations respect the Schouten bracket

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and $r \in \bigwedge^2 \mathfrak{g}$. For $x \in G$ let $\lambda_x$ denote the left multiplication by $x$. Let $[\cdot, \cdot]$ ...

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### Faithful locally free circle actions on a torus must be free?

Do we have an example of a smooth action $S^1 \curvearrowright T^n$ which is faithful, locally free but not free?
I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$.
Another ...

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### Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made.
In this paper, $\mathfrak{g}$ is a ...

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### Does every nilpotent orbit have an element supported on the simple root spaces?

Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha_1, \dots ,\...

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### Summing over roots of a simple Lie algebra and Deligne series

For a simple Lie algebra $\mathfrak{g}$ we can define a Killing form $K(X,Y) \equiv \frac{1}{2 h^\vee}\operatorname{tr}(\mathfrak{ad}_X \mathfrak{ad}_Y)$, where $\mathfrak{ad}_X Y \equiv [X, Y]$ as ...

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### Almost conjugate subgroups of compact simple Lie groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition:
Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...

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### A question from the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel

In the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel, the author has made the following statement in the proof of Corollary D in page no. 18 ( https://arxiv.org/pdf/...

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### Centralizer of a reductive subgroup

Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...

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### Choice of generators to make the centralisers connected

In $G=\operatorname{PGL}_{2n}(\textbf{C})$, WLG, we assume all the toral elementary abelian 2-subgroups in discussion are in $T$, the image in $G$ of the group of diagonal matrices in $\operatorname{...

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### Finitely many Lie primitive subgroups

A closed subgroup $ \Gamma $ of a Lie group $ G $ is called Lie primitive if it is not contained in any proper positive dimensional closed subgroup.
A closed subgroup not contained in any positive ...

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### Lattices in $p$-adic groups

What are the examples of lattices in $\operatorname{SL}_n(\mathbb{Q}_p)$ with $n\geq 3$ or in other semisimple $p$-adic groups of higher rank?
It is known $\operatorname{SO}_n(\mathbb{Z}[1/p])$ is a ...

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### Problem in understanding the proof of cocycle condition for cocommutator

Let $G$ be a Poisson–Lie group with Poisson bivector field $\pi$. Let $\pi^{R} \colon G \longrightarrow \bigwedge^2 \mathfrak{g}$ be defined by $$\pi^R (x) = (d_x R_{x^{-1}} \otimes d_x R_{x^{-1}}) \...

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### Lie algebra-Lie group correspondence of an exact sequence

Is there, for any given sequence $\mathfrak{g}_1\xrightarrow{f_1}\mathfrak{g}_2\xrightarrow{f_2}\mathfrak{g}_3$ of finite-dimensional Lie algebras over $\mathbb{R}$ with $\mathrm{Im}\,f_1 = \mathrm{...

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### The tangent bundle of $G \times_H M$

Let $G$ be a Lie group with a closed subgroup $H$, and let $M$ be a smooth $H$-manifold. I am searching for a reference where it is proved that the tangent bundle of $G \times_H M$ is isomorphic to ...

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### Algorithm for finding the symmetries of a linear operator

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Let $V, W$ be finite dimensional complex vector spaces and $M\in \Hom(V, W)$ a full rank linear map. I want to see if there exists a Lie group ...

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### Equivalence between the existence of a nonempty open set of elliptic elements and a compact Cartan subgroup

In Goldman's book on Complex Hyperbolic Geometry, on page 203, it is stated that for a real semisimple Lie group $G$, the following are equivalent:
$G$ contains a nonempty open subset of elliptic ...

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### Fourier transform of functions mapping manifolds, is there a definition?

$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form
$$
f : \mathbb{R} \to \SO(3)^n
$$
Since $\SO(3)$ is a compact group so is $\SO(3)^n$.
Now if ...

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### Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ...

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### Non-invariant forms on loop Lie algebra of semisimple Lie group

Let us consider a Lie group $G$ with Lie algebra $\mathfrak{g}$ and let $L\mathfrak{g} = C^\infty(S^1, \mathfrak{g})$ the Lie algebra of the loop group $LG$.
My question is about continuous Lie ...

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### Normalizer of maximal torus is maximal in compact simple group

This is a cross post from MSE of
https://math.stackexchange.com/questions/4562196/normalizer-of-maximal-torus-is-maximal
Let $ T $ be a maximal torus in a compact connected simple Lie group $ K $. For ...

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### Why is $\mathrm{SO}(4)$ not a simple Lie group?

$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ ...

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### Parallelizability of Lie monoids

A Lie monoid is a monoid, together with a structure of a smooth manifold (possibly with a boundary), such that the monoid multiplication is smooth.
If all left (or right) translations in a Lie monoid $...

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### Tensor product of fundamental representations

Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-...

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### Does elimination imply projection of symmetry?

Let suppose we have system of ODE (linear for simplicity) for two unknown functions $x(t), y(t)$
$$
x''(t) = a(t) x'(t) + b(t) y'(t) + c(t) x(t) + d(t) y(t),
$$
and
$$
y''(t) = e(t) x'(t) + g(t) y'(t) ...

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### Fourier transform and rotations in 3d

Let $f\in \mathcal{S}(\mathbb{R}^3)$ be a Schwartz function invariant under rotations in $\mathbb{R}^3$ and let $\hat f\in \mathcal{S}(\mathbb{R}^3)$ be its Fourier transform, i.e. $\hat f(p)=\int_{\...

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### Positive radial function with compactly supported Fourier transform

Let $G$ be a non-compact semisimple Lie group with finite center (for example $SL_2(\mathbb{R})$). Choose a maximal compact subgroup $K$. A bi-$K$-invariant function is called radial. Let $A$ be a ...

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### On the center of the universal enveloping algebra of a Lie algebra

Consider a finite-dimensional Lie algebra over C. Sometimes (e.g. for any semisimple Lie algebra) the center of its universal enveloping algebra is isomorphic to a freely generated polynomial ring.
...

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### Symmetric spaces and spherical functions

Let $G$ be a connected compact Lie group, $\sigma$ an involutive automorphism and $K$ a subgroup such that $(G^\sigma)_0\subset K\subset G^\sigma$. Then $M=G/K$ is a Riemannian symmetric space.
...

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### Relation between projective representation and the representation of the universal cover of a Lie Group

I am reading this paper, in what says exactly:
"Weare dealing with a ray representation os the conformal group AND THEREFORE with a representation of the universal covering group of the conformal ...

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### Injectivity of the cohomology map induced by some projection map

Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...

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### Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds

I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...

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### How many diagrams interlace a given Young diagram?

For a fixed partition $\lambda=(\lambda_1\geq\dots\geq \lambda_n)$ we say $\mu=(\mu_1\geq \dots \geq \mu_{n-1})$ $\textit{interlaces}$ $\lambda$ iff
$$\lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \...

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### Derived subgroups of 2-adic congruence subgroups of $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be a prime, and let $\Gamma_r$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^r\mathbb{Z}_p)$.
Explicit formulas with formal ...

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### Free $S^1$-action on compact homogeneous spaces

Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.
If $r(G) > r(K)$ (...

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### Question about coadjoint orbits of compact connected Lie groups

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Denote by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $r$ be an element of $\mathfrak{g}^*$ such that $G_r$ the stabilizer of ...

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### Generalization of $G/T \simeq G_\mathbb{C}/B$

Let $G$ be a compact Lie group and Let $G_\mathbb{C}$ be its complexification. Let $T$ be a maximal torus of $G$ and let $X$ be the quotient $G/T$.
Consider $H$ to be a Lie subgroup of $G$ and denote ...

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### Maximum symmetry metric on irreducible compact symmetric space

Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...