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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Is the homogeneous coordinate ring of a flag variety a UFD?

I was wondering if $G$ is a semisimple complex algebraic group, then is the homogeneous coordinate ring of a flag variety a UFD or not?
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Extension of an involution on $G$ to an involution on $G_\mathbb{C}$

I asked this question on MSE https://math.stackexchange.com/questions/4475382/extension-of-an-involution-on-g-to-an-involution-on-g-mathbbc but didn't receive any answer so far. My question is the ...
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Right-invariant metrics on the unitary groups and embeddings in the complexification

Let $G = SU(n)$ and $G_c = SL(n, \mathbb{C})$. Let $g$ be a right-invariant metric on $G$ and let $g_k$ be the Killing metric on $G_c$. Define the map $p$ from $G_c$ to $G$ which maps $h \in G_c$ to $$...
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What are the maximal closed subgroups of $ SU_3 $?

What are the maximal closed subgroups of $ SU_3 $? This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that question, treating only the case of connected subgroups, are ...
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Question on a remark in Speh's paper

I am reading Birgit Speh's paper entitled "Unitary representations of Gl(n,R) with nontrivial (g,K)-cohomology" in Invent. Math. 71 (1983), no. 3, 443–465. In Remark 1.2.2.(b), it says that &...
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What is the importance of Cartan decomposition of a semi-simple Lie algebra?

I just started learning about Cartan decomposition of semi-simple Lie algebras, and I'm curious to know what are their applications in studying semi-simple Lie algebras. My guess was that it might be ...
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Invariant sets in vector spaces

Given a real vector space $V$ and the set of all its linear operators $\mathcal{B}(V)$, I am wondering if there is way to determine, for a subset $X\subset V$, the collection of operators $G_X\subset \...
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Question about finite dimensional representations of a semi-simple Lie group

I have posted a question in MSE https://math.stackexchange.com/questions/4468138/question-about-finite-dimensional-representations-of-a-semi-simple-lie-group but didn't receive any comment or answer. ...
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Question about the inverse operator on PSL(2,R) with respect to Liouville measure

In GTM 259 chapter 9 and Katok Hasselbaltt Introduction to Modern Theory of Dynamical System chapter 5 (using the Iwasawa KAN decomposition) we see the Unit Tangent bundle of Hyperbolic half plane is ...
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When is the zero weight space of an irreducible $\frak{sl}_{n+1}$-module non-trivial?

Take the complex semisimple Lie algebra $\frak{sl}_{n+1}$, with space of dominant integral weights $P(\frak{sl}_{n+1})$. For $V(\lambda)$ the irreducible representation corresponding to $\lambda \in P(...
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Asymptotic for spectral gap for irreps

Let G be a compact connected Lie group (nonabelian) and $(A,B)$ be a fixed pair of topological generators. Let $T: G\rightarrow U(d)$ be an irrep of dimension $d>1$. Then $|1+T(A)+T(B)|<3$, ...
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The Weyl dimension formula for fundamental weights

The Weyl dimension formula is an equation to calculate the dimension of a simple $\frak{g}$-module $V_{\lambda}$, of highest weight $\lambda$, for $\frak{g}$ a complex semisimple Lie algebra. ...
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The geometry of the group of automorphisms of a manifold

Given a manifold $M$, the group $Aut(M)$ is made of diffeomorphisms $M\to M$. Since the complete vector fields on $M$ form an infinite dimensional Lie algebra, and each generates a 1 dimensional Lie ...
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Identifying the conformal equivalence class of a 2-torus subgroup of the cubical 3-torus

Let K, L, M be integers with gcd(K,L,M) = 1. They determine a connected Lie subgroup G = G(K,L,M) of the cubical 3-torus (ℝ/ℤ)3 via G = {(x,y,z) ∊ (ℝ/ℤ)3 | Kx + Ly + Mz = 0} (where 0 denotes the ...
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Character of the Young product of representations of a Lie group

For a compact (reductive/semisimple) Lie group $G$ with a maximal torus $T$, which I will identify with a subgroup of ${\mathbb{C}^*}^n$ (for simplicity let's just say that $G\leqslant GL(n,\mathbb{C})...
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Ree groups and Moufang octagons

Consider a Ree group of type $^2\mathrm{F}_4$, defined over the field $k$. Tits showed that every Moufang generalized octagon arises as a natural geometric module on which a Ree group of this type ...
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Can the Lie group $\text{Aff}(1)$ be extended?

Translations over $\mathbb{R}^1$ (ie. $(x\rightarrow x+b)$) form the Lie group $\mathbb{R}^{+}$. If we add the scaling operations over $\mathbb{R}^1$ , we can form the Lie group $\text{Aff}(1)$, ...
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How to construct lattice points in bounded symmetric domain?

Consider the Hermitian bounded symmetric domain for $k \leq m$: $$ C_{k, m} = \{ Z \in \mathbb{C}^{m\times k} \,|\, Z^*Z < I_k \} $$ where $I_k$ is the $k\times k$ unit matrix. If I am not mistaken,...
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Is a fixed subgroup of a compact Lie group cotorally included in finitely many conjugacy classes?

Let $G$ be a compact Lie group, in the next discussion we consider only its closed subgroups without specifying it. We say that a subgroup $L$ is a cotoral subgroup of $K\leq G$ if $L \trianglelefteq ...
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Connection between Grothendieck's homotopy hypothesis and Lie's second and third theorems?

I'm not an expert on homotopy theory, but I speculated about this in my thesis, so I figured I'd ask about it here. As I understand it, the homotopy hypothesis says that $\infty$-groupoids, with $\...
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Example on pseudo-groups

Definition: ‎A pseudo-group is a collection $\mathcal{G}$ of (locally defined) invertible smooth diffeomorphisms of a manifold $M$. ‎The simplest example of a pseudo-group is the collection of all ...
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Centralizer of an element in a matrix Lie group whose Jordan form is given

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}$Let $G\subset \GL_n(\mathbb{C})$ be a complex matrix Lie group, ...
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Closedness of subgroup corresponding to semi-simple real Lie subalgebra

I have an impression (but could be wrong) that I heard that for any semi-simple (real) Lie subalgebra $\mathfrak{k}$ of $\mathfrak{gl}(n,\mathbb{R})$ there exists a connected closed Lie subgroup $K\...
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On fixed point sets of actions of compact Lie groups

Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$. Is it true ...
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What does the boundary of convex hulls look like in matrix Lie groups?

Let $G$ be a compact matrix Lie group under the Killing form metric $\langle \xi, \eta \rangle_g = -\frac{1}{2}\text{tr}((g^{-1}\xi)^T(g^{-1}\eta))$ for $g \in G$ and $\xi,\eta \in T_gG$. Let $C \...
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6 votes
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Algorithmically handling the Spin groups in larg(ish) dimensions

Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$? By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute ...
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Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group?

Let $G$ be a semisimple (not just reductive) group over a field $k$. I believe that the question I am asking is what was meant in the second paragraph of Tits building of a linear algebraic group. I ...
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Hausdorff distance in compact Lie groups

Let $G$ be a compact Lie group with a compatible biinvariant metric $d$. The hyperspace $K(G)$ of nonempty compact subsets of $G$ is a compact metric space with the Hausdorff metric, and it is easy to ...
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Decomposition about splitting of symmetric spaces of compact type

I get stuck in the following question: Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...
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Questions about symmetric spaces

I'm a little confused with the following questions: (1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$? (2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...
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3 answers
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Conjugacy classes in the automorphism group of a simple Lie algebra

A lower bound of the number of conjugacy classes in the automorphism group of a simple Lie algebra $\mathfrak{s}$, of finite dimension over an arbitrary field $\mathbb{F}$, can be the size of the ...
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2 votes
1 answer
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Non-isomorphic direct products of a solvable and a semisimple Lie algebra

Given a solvable Lie algebra $\frak{a}$ and a semisimple Lie algebra $\frak{g}$ we can take their semidirect product $\frak{a} \rtimes \frak{g}$, with respect to a Lie algebra map $\frak{g} \to \...
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Fusion rules for the Lie algebra $\frak{so}_{2n+1}$

For the Lie algebra $\mathfrak{so}_{2n+1}$ where can I find a description of the fusion rules of it fundamental representations? In more detail: For $\pi_i$ and $\pi_j$ two fundamental weights of $\...
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1 vote
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Homogeneous metrics on compact Lie groups

Given a differentiable manifold $M$, a Riemannian metric $g$ on $M$ is called homogeneous if its isometry group acts transitively on $M$. In that case, given any group $G$ acting transtitively on $(M,...
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Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
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Compact Lie groups are rational homotopy equivalent to a product of spheres

According to [1] and [2], it is “well-known” that a compact Lie group $G$ has the same rational homology, and according to [2] is even rational homotopy equivalent, to the product $\mathbb{S}^{2m_1+1} ...
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Minimal dimension for $ \mathrm{PSU}_n $ as a matrix group

$\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$ Here's the new question: $ \SU_2 $ is a subgroup of $ \GL_2(\...
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4 votes
1 answer
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Aschbacher classes for compact simple group

Posted this to MSE several weeks ago and it got 3 upvotes but no answers or even comments so I'm cross-posting to MO Aschbacher's theorem says that every maximal subgroup of a finite simple classical ...
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2 votes
1 answer
216 views

Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely

Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...
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7 votes
2 answers
533 views

Question on KAK decomposition

Let $G$ be a semisimple Lie group and let $G = KAK$ be a Cartan decomposition. For $\mathrm{SL}_2(\mathbb{R})$ it holds for every $g \in G$ that $KgK = Kg^{-1}K$. Does the same hold for every ...
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12 votes
1 answer
280 views

Mathematical explanation of orbital shell sizes: why is it sufficient to consider single-electron wave functions?

Motivation The question "Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?" asks for an explanation of the sequence 2, 8, 8, ...
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2 votes
1 answer
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Compact Lie groups as quotients of torsion-free compact metrizable groups

The question: (1) Is every compact Lie group $G$ isomorphic (as a topological group) to some quotient $H/N$ where $H$ is a torsion-free compact metrizable group? Or equivalently: (2) Is every compact ...
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2 votes
1 answer
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Does the set of matrices with bounded recursive products form a fractal?

We are given three matrices $A,B,C$ which have determinant 1, are not unitary and that their size is the same. Consider the following process. On each step we take three words $W_1,W_2,W_3$ consisting ...
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7 votes
2 answers
384 views

Quadratic forms on $\mathbb{R}^{16}$ coming from octonions

$\DeclareMathOperator\RRe{Re}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sym{Sym}$Let $\mathcal{H}_2(\mathbb{O})$ denote the (10-dimensional) real vector space of octonionic Hermitian matrices ...
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The closure of the subgroup generated by a vector field may not be compact

Suppose $X$ is a vector field on a manifold $M$, consider the one parameter group: $$L=\left\{\phi^t_X: t\in\mathbb{R}\right\}$$ where $\phi^t_X$ is the flow of the vector field $X$, which sends $p\in ...
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6 votes
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Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups

If $G$ is a classical semisimple algebraic/Lie group over an algebraically closed field (maybe just say $\mathbb{C}$), viꝫ. $\mathit{SL}_n$, $\mathit{SO}_n$, $\mathit{Sp}_n$ (isogenies irrelevant here)...
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2 votes
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Element conjugate to a maximal torus

It is well known that any element in a compact connected Lie group is conjugate to an element in a maximal torus. Let G be a Lie group that is not necessarily compact and/or connected. Let $x\in G$ ...
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48 votes
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Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?

$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
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1 vote
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A basic application of Mackey's theorem

Let $G=GL(2,\mathbb {F}_q)$ and $B=\left\{\begin{pmatrix} * & * \\ 0 & * \end{pmatrix}\right\}$ the Borel subgroup of upper triangular matrices. Let $\chi_1$ and $\chi_2$ be the characters ...
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2 votes
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Question about adjoint orbits

I am looking for a proof or a reference of the following claim: Let $G$ be a real connected semi-simple Lie group and let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be the Cartan decomposition for its ...
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