# 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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### Hilbert representation with compactness and *-property

Given a continuous representation $\pi$ of a Lie group $G$ on a Hilbert space $H$, not unitary.
Suppose that
it is a $\star$-representation for the Lie algebra, i.e., $\pi(X)^\star=\pi(-X)$ on the ...

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### Unitary representations of $\mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{C})$

Does the left, unitary action of $\mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{C})$ on $\mathcal{L}^2(\text{H}^{+}_{3})$ integrate to its Lie group, i.e $\text{SL}(2,\mathbb{C})\times \...

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### Lie group framing and framed bordism

What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...

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### Are all infinite-dimensional Lie groups noncompact?

Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?

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### Holomorphic discrete series vs. discrete series

(I apologize in advance if this question is too naive for experts.) Let $G$ be a real semisimple Lie group. I know that holomorphic discrete series representations are only a part of all the discrete ...

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### What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?

Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...

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### Gauge Lie groupoid associated to $SO(3)$ double cover

From each Lie group $G$ and principal $G$-bundle $P \rightarrow E$ one can form an associated (or gauge) Lie groupoid as the quotient of pair groupoid by the action of $G$ on $P \times P$
$$ \frac{P \...

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### Find good representatives for something related to the orbit of $\mathrm{GL}_n\times \mathrm{GL}_m$ acts on $M_{n,m}$

$\DeclareMathOperator\GL{GL}$Fix a field $F$, and we only talk about spaces and groups over it. We consider the $n\times m$ matrix space denoted by $M$. And $\GL_n \times \GL_m$ acts canonically on $M$...

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### Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?

In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...

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### On Haar measure and Spherical measure [closed]

Let $d$-dimensional complex sphere be
$$\{(c_1,\cdots,c_{d})\sum_{i=1}^{d} |c_i|^2=1.\}$$
We can define the Haar measure on this sphere by regarding the unitary group $U(d)$.
We can regard the $d$-...

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### Examples of Lie groups where $G\to G/H$ splits topologically but not as groups

Let $G$ be a connected Lie group and $H$ a normal connected subgroup. I'm looking for examples where $G\to G/H$ has a section as topological spaces, but not as Lie groups.
My motivation here is to ...

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### What's the point of geometric representation theory?

Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...

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### Representation theory of $\mathrm{GL}_n(\mathbb{Z})$

I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...

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### Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces

For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces?
For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...

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### Generators of simple Lie groups and finite word length

Let $G$ be a connected simple Lie group with finite center. Let $a=\mathrm{exp}(X)$ be a semisimple element. Then we can decompose the lie algebra of $G$ into the direct sum of the eigenspaces of $\...

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### Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$

In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...

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### Relation between real forms of Lie algebras and root systems on pseudoeuclidean vector spaces

This might be trivial but I cannot see it clearly.
Simple complex Lie algebras are fully classified by the root systems arising from the Cartan subalgebra for which the Euclidean norm is the Cartan-...

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### Triple product formula on $K = \mathrm{SU}(2)$

Let $K = \mathrm{SU}(2) = \{ k[\alpha ,\beta] \mid \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with
$$ k [ \alpha , \beta ] =
\begin{pmatrix}
\alpha & \beta \\
- \...

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### On the existence of non-arithmetic lattices in algebraic groups over $\mathbb{Q}$

$\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator\PU{PU}$Let $G$ be a simple algebraic group over $\Q$ such that $G(\R) \simeq \prod_i G_i$, with each $G_i$ being the Lie ...

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### Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?

Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...

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### Efficient decoding of the E8/Leech lattice

Background:
Our goal is to quantize a sequence of floating point numbers generated i.i.d. from a standard Gaussian source and minimize the MSE reconstruction error. We can use two bits for each sample....

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### Closed subgroups in Ratner's orbit closure theorem on unipotent flows

Let $G$ be a semisimple (real or $p$-adic) Lie group and $\Gamma$ a discrete and cocompact subgroup of $G$, as in the setting of Ratner's theorems on unipotent flows (see for example here \url{https://...

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### The adjoint representation of a Lie group

Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...

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### $\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$

Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an ...

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### The basic representation of $LU(1)$

Let $H = L^2(U(1),\mathbb{C})$. The "basic" irreducible projective level 1 representation $\mathcal{H}$ of the loop group $LU(1)$ has underlying Hilbert space isomorphic to $\smash{\hat{\...

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### Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\...

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### Branching problem of representation of Lie groups and orbit method

Branching problem asks how a restriction of an irreducible representation of $G$ to a subgroup $H$ decomposes. In case of (real) Lie groups, after labeling irreducible representations via highest ...

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### Is the inclusion of the maximal torus in a simply connected compact Lie group null-homotopic?

Let $G$ be a simply connected compact Lie group and $T$ its maximal torus with inclusion $i:T \hookrightarrow G$.
By simply connectedness of the group $G$ and asphericity of the torus $T$, the induced ...

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### Heat kernel of left-invariant metric on 3-sphere

This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...

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### Relation of geometric and polyhedral convergence

By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...

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### Large deviations for geodesic flows of Lie groups

Let $G$ be a Lie group with finite Haar measure $\mu$. Choose $g\in G$ uniformly according to $\mu$. Let $\Phi^s$ denote the geodesic flow. Has the following been studied
$$\lim_{t\to\infty}\mu\left(\...

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### Expected value of the length of the shortest non-zero vector in a lattice?

$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...

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### Boundedness of dimension of representations that restrict to a fixed representation of a normal subgroup

Let $G$ be a compact Lie group, $H$ a closed subgroup and $W$ an irreducible real representation of $G$. Then it follows from Frobenius reciprocity and Bott’s definition of induced representation that ...

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### Where to find English translation of Pansu's paper from Ann. Math?

Where can I find English translation of the following paper?
P. Pansu,
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French. English summary) [Carnot-...

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### Gradient descent under the presence of symmetries

Let $M$ be a Riemannian manifold (I'm happy to assume it is Euclidean space) with a function $f: M \to \mathbb R$ and a group of isometries $G$ acting on $M$ and preserving $f$, i.e., $f(gm) = f(m)$ ...

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### When does a group act effectively and holomorphically on some Riemann surface?

Given a Riemann surface $X$, we have some fairly standard methods for identifying which groups $G$ admit an effective and holomorphic action $G \times X \to X$. For instance, some fairly elementary ...

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### Division of fibration by $\Sigma_{n}$ gives Serre fibration

This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...

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### Are there always flat connections?

Let $G$ be a simply connected Lie group and $\Gamma$ a cocompact discrete and torsion-free subgroup. Is there a (real or complex) smooth vector bundle $E$ over the manifold $G/\Gamma$, which does not ...

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### Centers of universal enveloping algebra of complex Lie algebras

Let $\mathfrak{g}$ and $\mathfrak{g'}$ be complex Lie algebras such that $\mathfrak{g}$ is a subalgebra of $\mathfrak{g'}$. Let $Z(\mathfrak{g})$ and $Z(\mathfrak{g'})$ be the centers of the universal ...

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### Perpendicular mapping from one matrix group to a closed matrix subgroup

Let H be a connected closed Lie subgroup of G, which is a connected closed Lie subgroup of GL(n,𝔽), where 𝔽 = ℝ or 𝔽 = ℂ. Assume that G and H are each endowed with the riemannian metric inherited ...

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### Homogeneous metric connections on 3-dimensional Lie groups

Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...

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### The space of ergodic elements of a topological or Lie group

Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...

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### Rotation number for homeomorphisms of a Lie group other than $S^1$

Let $G$ be a Lie group whose Lie algebra is $\mathfrak{g}$ with exponential map $\exp:\mathfrak{g}\to G$.
For what kind of Lie group $G$ the standard process of definition of rotation number ...

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### Why can't a Lie group act transitively on a finite volume hyperbolic manifold?

In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?",
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...

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### Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?

Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...

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### Do the exceptional root systems arise in the real world?

I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...

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### Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression

From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...

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### Adjoint action on orthogonal complement

Consider a compact Lie algebra $\mathfrak{g} \subset \mathfrak{u}(n)$ and its associated connected, compact Lie group $G$. Let $\mathfrak{g}^{\perp}$ denote $\mathfrak{g}$'s orthogonal complement (as ...

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### Algorithmic representation of the Spin (and Pin) group [duplicate]

Performing algorithmic computations in $\mathit{SO}_n(\mathbb{R})$ or $\mathit{O}_n(\mathbb{R})$ is easy: its elements are represented by $n\times n$ orthogonal matrices of reals so, assuming we have ...

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### Reference for group-algebra/exp-log like identites in combinatorics

I've encountered several identities in combinatorics that resemble inversion formulas, as shown below,
Here, $f_i, g_k$, $\forall i,k \in \mathbb{N}$, are coefficients of some formal power series.
I ...