Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
2,363
questions
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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 ...
7
votes
1answer
288 views
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 ...
6
votes
1answer
175 views
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 ...
2
votes
0answers
104 views
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, ...
1
vote
0answers
118 views
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 ...
0
votes
0answers
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 $X \subset {\cal M}$? ...
3
votes
2answers
199 views
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 ...
1
vote
1answer
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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124 views
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}_{...
2
votes
0answers
89 views
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? ...
1
vote
1answer
64 views
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 ...
8
votes
3answers
265 views
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, ...
4
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141 views
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\|}...
2
votes
0answers
48 views
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?
4
votes
0answers
170 views
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 ...
9
votes
1answer
319 views
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....
7
votes
1answer
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 $\...
2
votes
0answers
83 views
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 ...
3
votes
0answers
87 views
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, \...
4
votes
1answer
167 views
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 ...
4
votes
0answers
102 views
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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votes
0answers
110 views
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 $\mathcal{G}$. ...
9
votes
1answer
265 views
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 ...
2
votes
1answer
129 views
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 "...
2
votes
0answers
34 views
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 ...
5
votes
1answer
313 views
$\pi_{2n-1}(\operatorname{SO}(2n))$ 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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0answers
54 views
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
$$\{\sigma^j \times 1_2, \quad \quad \sigma^i \times \sigma^j, \quad \quad 1_2 \times \sigma^j\},...
3
votes
2answers
237 views
Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
1
vote
1answer
259 views
What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(...
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votes
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28 views
Generating $K$-types of a $(\mathfrak g,K)$-module for $K$ disconnected
Let $G$ be a real reductive Lie group, let $K$ be a maximal compact subgroup of $G$, and let $V$ be a $(\mathfrak g,K)$-module. For $\sigma\in\widehat{K}$ we denote the $\sigma$-isotypic component of $...
4
votes
2answers
174 views
Schur positivity of a polynomial
Suppose a polynomial of the form
$$\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 ...
4
votes
0answers
109 views
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}$, ...
3
votes
0answers
101 views
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 ...
2
votes
0answers
133 views
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)$ ...
2
votes
1answer
287 views
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 ...
3
votes
1answer
123 views
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 ...
6
votes
3answers
337 views
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 ...
3
votes
2answers
78 views
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 ...
3
votes
0answers
102 views
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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vote
0answers
123 views
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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votes
0answers
435 views
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 ...
1
vote
1answer
122 views
$G/T$ 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 ...
13
votes
3answers
460 views
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 ...
2
votes
1answer
119 views
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 ...
10
votes
2answers
361 views
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-...
1
vote
0answers
101 views
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(\...
4
votes
2answers
225 views
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 ...
2
votes
0answers
86 views
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 ...
6
votes
2answers
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 ...
3
votes
0answers
95 views
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 ...
