# 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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### Integral functional on algebraic varieties

Suppose that $X$ is a smooth complex algebraic variety and $X_{\mathbb{R}}$ is a real form of $X$. If $X_{\mathbb{R}}$ is compact and oriented as a real manifold, then it will admit a natural ...

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### Homotopically equivalent compact Lie groups are diffeomorphic

I have the following conjecture:
Two homotopically equivalent compact Lie groups will be diffeomorphic. It may be necessary to restrict ourselves to only semisimple Lie groups. For simply connected ...

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### Viewing exceptional Lie algebras via the classical ones

I've been trying to understand the exceptional Lie algebras through the classical ones that I am more familiar with. In particular I wanted to get a handle on the root spaces and most discussions that ...

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### Erlangen program for “network geometry”

The subject of network geometry (Boguna et al., Network Geometry,
Nature Reviews Physics 2021) looks at "geometric aspects" of complex networks.
This is about studying a metric on the nodes, ...

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### Coadjoint orbits

I've posted the following question some days ago in math.stackexchange https://math.stackexchange.com/questions/4155747/co-adjoint-orbit but I didn't get any answer!
While I was trying to teach my ...

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76 views

### Cohomology group of a submanifold or Lie subgroup

In general: if one knows the cohomology group of some manifold ${\cal M}$, i.e. $H^n ({\cal M})$, are there known results for the same cohomology group $H^n (X)$ of a submanifold $X \subset {\cal M}$? ...

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### Three dimensional real Lie groups with cocompact discrete subgroups

I would like to know what are all the real three dimensional Lie groups (simply connected) that can act transitively and locally freely on a compact three dimensional manifold?
This is equivalent to ...

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156 views

### Classification of the group action

Let $G$ be a closed subgroup of $O(n)$ such that $\mathbb R^n/G$ is isometric to $\mathbb R^{n-2} \times \mathbb R_+$. Can we have a classification of $G$ up to conjugation?

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### Proving that $\mathrm{SL}_2(\mathbb{Z}[i])$ is a lattice in $\mathrm{SL}_2(\mathbb{C})$

I need to show that $\mathrm{SL}_2(\mathbb{Z}[i])$ is a lattice in $\mathrm{SL}_2(\mathbb{C})$. I was thinking of applying Borel-Harish Chandra theorem, which says that:
Let $G \subseteq \mathrm{GL}_{...

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### Fundamental weights for $\mathrm{PGL}_n(\mathbb{C})$

I'm interested in the fundamental weights of non simply-connected simple Lie groups, and more specifically those of $\mathrm{PGL}_n(\mathbb{C})$.
Is there a general method to find all the generators? ...

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### Cosemisimple pointed Hopf algebras

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Every cosemisimple pointed Hopf $\mathbb{K}$-algebra $A$ is easily seen to be cocommutative. Does this imply that $A$ is the ...

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### Mechanical systems with their configuration space being a Lie group

Cross-posted from Physics.SE
In Marsden, Ratiu - Introduction To Mechanics And Symmetry there is a certain focus on reducing cotangent bundles of Lie groups. More precisely, if $G$ is a Lie group, ...

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### Zariski density for certain subsemigroups

$\DeclareMathOperator\GL{GL}$Suppose that we have a Zariski-dense subgroup $\Gamma$ of $\GL(d,\mathbb{R})$. Let $\delta$ be the abscissa of convergence of the series
$$
\sum_{x \in \Gamma} e^{-s \|x\|}...

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### Are the odd dimensional spheres Poisson homogeneous spaces?

Are the odd dimensional spheres $S^{2n+1}$, for $n \in \mathbb{N}_{\geq 1}$, Poisson homogeneous spaces in the sense of Drinfeld?

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### Are there mathematical/physical applications of these Weyl equivariant maps?

Let $G$ be a compact Lie group and $T$ a choice of maximal torus. Denote the corresponding Lie algebras by $\mathfrak{g}$ and $\mathfrak{t}$. Elements of $\mathfrak{t} \otimes \mathbb{R}^3$ are called ...

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### A detail in the proof of Schur's lemma: the closures of the $\mathcal{Ker}$ and $\mathcal{Im}$ of the intertwiner

$\renewcommand\Im{\operatorname{\mathcal{Im}}}\newcommand\Ker{\operatorname{\mathcal{Ker}}}$I was sure that this is a trivial question and placed it on Math Stackexchange
https://math.stackexchange....

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187 views

### How special are homogeneous spaces?

Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$?
Related questions/approaches: Of course we need $\...

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### Is this equivariant function constant?

Let $G$ be a linear algebraic group (think of $SL_n(\mathbb{R})$), $B$ its Borel (standard minimal parabolic) subgroup (think of upper triangular subgroup), and let $\Gamma \leq G$ be a cocompact ...

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### Logarithm on formal power series continuous?

Denote $V:=\mathbb{R}^d$ and consider the Cartesian product $V^\infty:=\prod_{k=0}^\infty V^{\otimes k}$ together with its canonical projections $\pi_k : V^\infty\rightarrow V^{\otimes k}, (v_0, v_1, \...

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### Finite covolume of uniform lattice in quotient group

Let $G$ be a locally compact group, let $N \leq G$ be a (proper) closed normal subgroup and let $\Gamma \leq G$ be a uniform lattice, i.e., a discrete subgroup such that $G/\Gamma$ Is compact.
Suppose ...

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### Is there a smooth Weyl equivariant map from this quotient space into $G_2/T^2$?

It is known that $G_2$ acts transitively on $S^6$ with fibers $SU(3)$. Let us consider the following set $P$ of complex unitary $7 \times7$ matrices $A$, where
$$ A = (v_0, \, v_1, \, v_2, \, v_3, \, ...

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### density of unipotent flows in algebraic groups

Let $\mathcal{G}$ be a reductive algebraic group over $\mathbb{Q}$ with a model $G$ over $\mathbb{Z}$ such that $G(\mathbb{R})$ is compact modulo centre. Let $T$ be a maximal torus of $\mathcal{G}$. ...

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### The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$

I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...

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### A variation of closed-subgroup theorem

$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group.
I am pretty sure that this theorem should have a "...

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### Lie Groups and Lie algebras related to Jordan algebras

Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$.
In the paper "The Capelli Identity, Tube Domains, and the Generalized Laplace Transform"
Jacobson [J] has ...

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### $\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible for $n>1$?

Question: Is the element $\alpha$ in $\pi_{2n-1}(\operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?
My understanding so far —
An $\...

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### What subspace of $\operatorname{SU}(4)$ group keeps an element of the $\mathfrak{su}(2)$ subalgebra within $\mathfrak{su}(2)$ upon adjoint action?

Consider the Lie group $G_4=\operatorname{SU}(4)$ with (15) generators $T^a$. A basis for the latter is
$$\{\sigma^j \times 1_2, \quad \quad \sigma^i \times \sigma^j, \quad \quad 1_2 \times \sigma^j\},...

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### Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...

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### What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(...

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### Generating $K$-types of a $(\mathfrak g,K)$-module for $K$ disconnected

Let $G$ be a real reductive Lie group, let $K$ be a maximal compact subgroup of $G$, and let $V$ be a $(\mathfrak g,K)$-module. For $\sigma\in\widehat{K}$ we denote the $\sigma$-isotypic component of $...

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### Schur positivity of a polynomial

Suppose a polynomial of the form
$$\prod_i^d \sum_j^p x_i^{f_j}$$
clearly symmetric, where $f_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur ...

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### The coherence property of center of universal enveloping algebra for reductive Lie algebra?

Let $G' \subset G$ be two reductive Lie groups over $\mathbb{R}$ and $\mathfrak{g}_{\mathbb{C}}' \subset \mathfrak{g}_{\mathbb{C}}$ be their complexified reductive Lie algebra over $\mathbb{C}$, ...

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### Torsion-free subgroups in arbitrary lattices

Let $\Gamma $ be a lattice in a semi-simple Lie group $G$. If the Lie group $G$ is linear (that is, it has a faithful finite-dimensional linear representation), then $\Gamma$ contains a torsion-free ...

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### Spherical harmonics, $\frak{sl}_2$, and algebra gradings

Let $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $\operatorname{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $\operatorname{Pol}(S^2)$ ...

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### Is the representation of finite simple groups fully understood?

Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...

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### Rank of a Lie subgroup generated by two Lie subgroups

$\DeclareMathOperator\rank{rank}$Let $G$ be a compact connected Lie group and $H$, $K$ be two closed connected subgroups. By Mikhail Borovoi's answer to In a compact lie group, can two closed ...

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### Root system of fixed point Lie sub-algebra

It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...

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### When representations of reductive Lie group in a Banach space and in its Garding space have the same length?

Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of ...

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### The product of $Z(\mathfrak{g})$-finite functions is also $Z(\mathfrak{g})$-finite?

Let $G$ be a classical group defined over $\mathbb{Q}$.
Let $\mathfrak{g}$ be the Lie algebra of $G(\mathbb{R})$ and $U(\mathfrak{g}_{\mathbb{C}})$ its universal enveloping algebra of $\mathfrak{g}_{\...

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### Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$

Given a group $G$, we denote the center Z$(G)$, we like to know the
automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences:
$$...

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### Geodesics on algebraic manifold

A nonsingular algebraic manifold is an immersed manifold (slightly different from the usual algebraic manifold which is required to be embedded) $M \subseteq \Bbb{R}^n$ (with induced topology) that is ...

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### $G/T$ has finitely many $G^\theta$ orbits

Let $G$ be a compact connected Lie group and T be it's maximal torus. Let $\theta: G \rightarrow G$ be an involution on $G$ and let $G^\theta = \lbrace g \in G , \theta(g)=g \rbrace $.
I'm looking for ...

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### Extending group actions to vector bundles

Let $G$ be a group acting on a manifold $M$. Suppose $V$ is a rank $n$ vector bundle on $M$.
Is there any obstruction to extending the action of $G$ to $V$? In how many ways can the action be extended ...

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### Properties of stabilizers of adjoint action general linear group

Let $G=GL(n,\mathbb{C})$ and let us consider $x \in GL(n,\mathbb{C})$. I'd like to know whether the following is true: the stabilizer for the conjugation action $C(x)$ is special in the sense that ...

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### A specific coset decomposition of $\mathrm{GL}_n(\mathbb{C})$

Disclaimer: I am a theoretical chemist (not a mathematician). I have tried asking this question at Math SE with no luck (https://math.stackexchange.com/questions/4080696/a-specific-coset-decomposition-...

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### What is the analogue of Leibniz's rule for universal enveloping algebra?

Let $G$ be a reductive group over $\mathbb{R}$ and $\mathfrak{g}$ its complexitied Lie algebra.
Let $U(\mathfrak{g})$ be the universal enveloping algebra and $Z(\mathfrak{g})$ is the center of $U(\...

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### Low dimensional integral cohomology of $BPSO(4n)$

Toda has calculated the $\mathbb{Z}/2$‐cohomology ring of $BPSO(4n+2)$, and also gave the simple exceptional calculation of the $\mathbb{Z}/2$‐cohomology of $BPSO(4)$, in
Hiroshi Toda, Cohomology of ...

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### Deduce Sheffer's classification of orthogonal polynomials of A-type 0

Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...

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484 views

### Finite subgroups of $\operatorname{U}(2)$

Famously, the finite subgroups of $\operatorname{SU}(2)$ admit an ADE classification.
Question. Is there a similar result for finite subgroups of $\operatorname{U}(2)$? Are they
classified? If this ...

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### About the representation ring of a compact group

A question stuck in my mind when I was reading the paper "The representation ring of a compact Lie group" by Segal. He says on page one that I confine myself to the case of a compact Lie ...