Questions tagged [lattices-in-lie-groups]

In mathematics, especially in geometry and group theory, a lattice in $\mathbb{R}^n$ is a subgroup of $\mathbb{R}^n$ which is isomorphic to $\mathbb{Z}^n$, and which spans the real vector space $\mathbb{R}^n$. In other words, for any basis of $\mathbb{R}^n$, the subgroup of all linear combinations with integer coefficients forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell.

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When is the quotient of $\operatorname{GL}_n(\mathbb{R})$ by a discrete subgroup compact?

My question is exactly that on the title. I'm interested in the action of some (discrete) subgroup $H$ on $\operatorname{GL}_n(\mathbb{R})$ by left multiplication. For example, $H$ can be $\...
0 votes
1 answer
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Example of difficult base for the lattice problems

I'm interesting in the lattice problems (the one used in the post quantum crypto), especially the shortest vector problem. I'm trying to understand why it is hard by finding an example, but I can't ...
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1 answer
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Lattice defined on poset vs. Lattice defined on group?

I've seen two different definitions of the term lattice, one is defined on poset, the other one is defined on group. I believe these two are fundamentally different mathematical objects. But I'm not a ...
1 vote
0 answers
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Lattice not contained in any connected subgroup is not contained in any positive dimensional subgroup

Let $ G $ be a simple Lie group and let $ \Gamma $ be a lattice in $ G $. If $ \Gamma $ is not contained in any connected subgroup of $ G $ does that imply that $ \Gamma $ is not contained in any ...
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1 answer
53 views

Lattice in noncompact simple group is Ad-irreducible

Is every lattice in a Lie group Ad-irreducible? No. This is false for $ G $ compact because any finite subgroup is a lattice. And it is certainly false if $ G $ is not simple since a group can only ...
6 votes
1 answer
814 views

Finite maximal closed subgroups of Lie groups

$\newcommand{\G}{\mathcal{G}} \newcommand{\K}{\mathcal{K}} \DeclareMathOperator\SU{SU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\SO{SO}$Let $\G$ be a Lie group. I am interested in finite ...
4 votes
1 answer
139 views

Is the fundamental group of a closed orientable hyperbolic $3$-manifold isomorphic to a non-discrete subgroup of $\mathrm{PSL}(2, \mathbb{C})$?

Consider the fundamental group $\pi_1(M)$ of a closed orientable hyperbolic $3$-manifold $M$. Certain identifications $\tilde{M} \approx \mathbb{H}^3 \approx \mathrm{PSL}(2, \mathbb{C}) / \mathrm{Stab}...
1 vote
1 answer
139 views

Hyperplanes and sublattices correspondence

I struggled to understand the first 3 lines of this paragraph (see linked image), can someone elaborate? Not sure what is the inverse image in $\mathbb{Z}^2$ and how to understand the corresponding ...
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31 views

What does it mean that the distribution of a variable is well-defined?

In the proof of a lemma in a paper, the authors say "Observe the distribution of $\vec{d}$ is well-defined." What does it mean mathematically? Here is the picture of the notation, lemma, and ...
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Example of a cocompact lattice in SL(2,R) [duplicate]

I would like to understand compact hyperbolic manifolds a bit better. Hyperbolic manifolds may be described as quotients of hyperbolic space by a torsion-free, discrete subgroup (lattice) of the ...
1 vote
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30 views

Thin Groups and Lie Primitive Groups

Let $ G(\mathbb{R}) $ be a noncompact Lie group which is the real points of an algebraic group. Let $ \Gamma $ be a closed subgroup of $ G $. $ \Gamma $ is called Lie primitive if it is not contained ...
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Malcev endomorphism rigidity theorem for 1-connected nilpotent real Lie group?

This question might be really stupid, but I couldn't figure out the answer. Let $G$ be a 1-connected real nilpotent Lie group, and $\Gamma$ be a lattice in $G$. By Malcev, every automorphism $f$ on $\...
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Action of $GL_2(\mathbb R)$ on the space of lattices

A lattice in $\mathbb C$ is an abelian subgroup of $\mathbb C$ isomorphic to $\mathbb Z^2$ that generates $\mathbb C$ over $\mathbb R$. Define on the space $M$ of lattices in $\mathbb C$ a structure ...
1 vote
1 answer
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Computing a minimal sublattice containing two other sublattices, GCD of a lattice

I've been trying to read a paper regarding the analogue of a GCD for lattices, but I'm not sure I understand how to decipher this notion. This is given in Section 3.1 when the author discusses the '...
2 votes
1 answer
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Can the intersection of two non-commensurable lattices be Zariski-dense?

Let $\Gamma_1$ and $\Gamma_2$ be lattices of a simple non-compact Lie group $G$. Suppose that $\Gamma_1\cap\Gamma_2$ is Zariski-dense in $G$. Can we conclude that $\Gamma_1\cap\Gamma_2$ is also a ...
1 vote
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Number of elements in $\text{SL}(d,\mathbb Z)$ with bounded $2$-norm

I wonder if there are good estimate for the number of elements in $\text{SL}(d,\mathbb Z)$ with $2$-norm bounded by $T$ as $T \to \infty$, namely $$\#\{g\in SL(d,\mathbb Z):\|g\|_2:=\sqrt{ \sum_{ij}g_{...
1 vote
1 answer
50 views

Self-normalizing and irreducible in the adjoint representation implies maximal

Let $ G $ be a connected Lie group and $ \Gamma $ a subgroup. Let $ \mathfrak{g} $ be the Lie algebra of $ G $ and $ ad: G \to GL(\mathfrak{g}) $ be the adjoint representation of $ G $ and let the $ ...
0 votes
1 answer
128 views

Finite simple group which is maximal closed subgroup of lie group

The icosahedral subgroup of $ G=PSU_2 \cong SO_3(\mathbb{R}) $ is a finite simple group which is also maximal closed in $ G $. Do other Lie groups admit maximal closed subgroups which are also finite ...
3 votes
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Does every effective $T^4$-action on a simply connected 6-manifold admit a free subtorus action?

All simply connected $6$-manifolds which admit effective $T^4$-actions have been classified by Hae Soo Oh (See Theorem 1.1). The classification shows that only the following simply connected $6$-...
7 votes
1 answer
354 views

Is $\operatorname{SL}(n,\mathbb{R})/\operatorname{SL}(n, \mathbb{Z})$ a Hausdorff space?

The special linear group $\operatorname{SL}(n,\mathbb{R})$ of degree $n$ over $\mathbb{R}$ is the set of $n \times n$ matrices with determinant $1$, with the group operations of ordinary matrix ...
6 votes
2 answers
216 views

Does a lattice in $SL_n(\mathbb R)$ which is contained in $SL_n(\mathbb Z)$ have finite index in $SL_n(\mathbb Z)$?

A lattice $H$ in a locally compact group $G$ is a discrete subgroup such that the coset space $G/H$ admits a finite $G$-invariant measure. I have read several places that any lattice H in $\...
1 vote
1 answer
120 views

Cocompact lattice has infinite commutator subgroup

Let $ G $ be a noncompact semisimple Lie group and $ \Gamma $ a cocompact lattice in $ G $. Is the commutator subgroup of $ \Gamma $ always infinite? EDIT (I deleted two sub-questions, also I'm adding ...
2 votes
2 answers
132 views

Why do the eigenvalues of the irreps. of $\mathfrak{sl}_3(\mathbb{C})$ differ by integral linear combinations of $L_i - L_j \in \mathfrak{h}^*$?

(Preamble) In the book Representation Theory A First Course (Fulton, Harris), there is the following claim in the page 165 (written as an observation) without a proof: The eigenvalues $\alpha$ ...
1 vote
1 answer
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how can you tell if two lattices are the same

I'm looking at three different definitions of the $E_8$ lattice : $G_a$ is the coding theory version $G_b$ is the root lattice version $G_c$ is the wiki version According to this old question the way ...
1 vote
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38 views

A lattice acts in a locally compact group

I'm taking a topics course and trying to fill in details from a lecture. I'm hoping someone can help fill things in or point to a resource where these basics are covered. We have a lattice $\Lambda &...
3 votes
1 answer
125 views

Realizing a group as a cocompact lattice

Let $ \Gamma $ be a finitely generated torsion free group. Does there always exist a Lie group $ G $ containing $ \Gamma $ (or really a subgroup isomorphic to $ \Gamma $) such that $ G/\Gamma $ is ...
1 vote
1 answer
61 views

Examples of uniform lattices - reference [closed]

Being interested in uniform lattices (that is, discrete co-compact subgroups) of connected Lie groups, I am searching for a literature with abundance of examples.
3 votes
1 answer
56 views

Stabilizer of probability measure on projective space

I'm reading notes on Lattices on Lie groups (in French: Réseaux des groupes de Lie) by Yves Benoist and I have some troubles to understand some aspects on the setting & proof of the Lemma 7.3 (due ...
3 votes
1 answer
103 views

An isomorphism of unimodular lattices in $\mathbb{R}^n$

In the proof that $SL_{n}(\mathbb{Z})$ is a lattice in $SL_{n}(\mathbb{R})$, the following isomorphism is used $$SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z}) \cong \{\text{unimodular lattices in } \mathbb{R}^...
2 votes
2 answers
149 views

Projection of a lattice need not be a lattice

I have the following problem: Let $L$ be a $\mathbb{R}^n$ lattice (that is a discrete closed $\mathbb{R}^n$ subgroup). Let $E$ be a vector subspace of $\mathbb{R}^n$ and consider $\pi$ to be ...
0 votes
1 answer
50 views

Find $\{a, b, c\} \in \Bbb{Z}$ that minimise $\left\|v_0 + a\cdot v_1 + b\cdot v_2 + c\cdot v_3 \right\|$ where $\{v_0, v_1, v_2, v_3\} \in \Bbb{R}^3$

I need find all pairwise shortest distance between any two atoms 3d tiled parallelepiped. I can calculated the exact solution for where $\{a, b, c\} \in \Bbb{R}$ and considering all 8 rounding ...
2 votes
0 answers
45 views

From Automorphism of code to automorphism of lattice

From a code, a lattice can be constructed using many methods. For codes over $\mathbb{F}_2$, there is the straight construction \begin{equation} \Lambda(C) := \{v/\sqrt{2} \ | \ v \in \mathbb{Z}^n ...
2 votes
0 answers
300 views

Cocharacter lattice and coroot lattice

Consider a simply-connected Lie group $G$ with the maximal torus $H$ and Lie algebras $\mathfrak{g}$, $\mathfrak{h}$ respectively. The exponential map is $\exp(2\pi i \cdot) : \mathfrak{g} \to G$. A ...
1 vote
0 answers
19 views

semicontinuity of a function on the space of grids

Let $X_d$ denote the space of unimodular lattices, equipped with the topology where the convergence is defined by the convergence of basis ($x_n\to x$ in $X_d$ $\iff$ there exist basis of $x_n$, say $(...
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0 answers
53 views

Understanding $X_n = \big\{ \text{lattices } Λ < \Bbb R^n: \operatorname{covol} Λ = 1\big\} \cong SL(n, \Bbb Z) \setminus SL(n, \Bbb R)$

$X_n = \big\{ \text{lattices } Λ < \Bbb R^n: \operatorname{covol} Λ = 1\big\} \cong SL(n, \Bbb Z) / \operatorname{SL}(n, \Bbb R)$, with isomorphism given by: $\Bbb Zv_1 + \Bbb Zv_2 + \cdots + \Bbb ...
1 vote
0 answers
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Does the product of two Schur functions always have a lattice structure with respect to the dominance order of partitions?

The product of two Schur functions can be decomposed into a linear combination of other Schur functions according to the Littlewood-Richardson rule. This is also how the irreducible representations in ...
2 votes
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Finding the closest point in a root lattice

Let $L_n$ be a crystallographic root lattice, embedded inside $\mathbb{R}^n$. This means that $L_n$ is the $\mathbb{Z}$-span of the simple roots $\alpha_1, \ldots, \alpha_n \in \mathbb{R}^n$, which ...
1 vote
1 answer
320 views

Verifying some facts about finite-volume hyperbolic manifolds

Let $M$ be a finite-volume hyperbolic manifold. If we chop off the cusps (are there only finitely many?) to get a compact surface with (finitely many?) boundary components, and then lift to the ...
2 votes
1 answer
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Example 15.9 in Loring Tu's An Introduction to manifolds

Example 15.9 is showed as follows in Loring Tu's An Introduction to manifolds: Example 15.9 (Lines with irrational slope in a torus). Let $G$ be the torus $\mathbb{R}^2/\mathbb{Z}^2$ and $L$ a line ...
5 votes
1 answer
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Properties of the Weyl vector $\rho = \frac{1}{2} \sum_{\alpha > 0} \alpha$

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$. The group has a maximal torus $T$ with Lie algebra $\mathfrak{t}$. Let $\rho = \frac{1}{2}\sum_{\alpha > 0} \alpha$ be the ...
1 vote
0 answers
38 views

How to find rational elements of order 3 in the Lie group $A_2$

I'm new to Lie theory and I'm trying to find all the rational elements of order 3 in the Lie group $A_2$. I found all the elements of order 3 contained in the fundamental region, which is defined by $...
0 votes
1 answer
139 views

Why $\mathbb Z(\sqrt2)$ is not a lattice?

The set $\mathbb Z(\sqrt2) = \{a + b\sqrt2 : a, b \in \mathbb Z\}$ is not a lattice, according to the book of Robeldo = because when you replace $a, b \in \mathbb Z$ by $a, b \in \mathbb R$ we do ...
3 votes
1 answer
392 views

Haar measure of $\operatorname{SL}_n(\mathbb{R})$ in terms of the product $K\times A\times N$

I'm misunderstanding a standard argument about the interpretation of the Haar measure on $K\times A\times N$. (See Lemma 2.5 on page 145 of Bekka-Mayer for example.) Let $G=\operatorname{SL}_n(\mathbb{...
0 votes
0 answers
22 views

How does $g\in GL(n,\mathbb R)$ affect the ratio $\lambda_i(gL)/\lambda_i(L)$ of a lattice $L$

Let $L\le \mathbb R^n$ be a lattice and $g\in GL(n,\mathbb R)$. Suppose $g$ is fixed but $L$ is allowed to vary. Let $\lambda_i(L)$ be the $i$-th minimum of the lattice $L$ for $1\le i \le n$. I ...
0 votes
1 answer
51 views

How does the action of $GL(n,\mathbb R)$ affect the $k$-th minimum of a lattice?

Let $\Lambda\le \mathbb R^n$ be a lattice and $\lambda_k(\Lambda):=$ the $k$-th minimum of the lattice ($k=1,\dots,n$), namely the $k$-th shortest distance from a lattice $\Lambda$ to the origin (...
1 vote
0 answers
68 views

Co-compact lattice in locally compact hausdorff groups

Bekka-Mayer in their book below, in II.2 says: If $\Gamma$ is a discrete and cocompact subgroup of a locally compact group $G$, then $\Gamma$ is a lattice in $G$. I can't seem to prove it. Of course ...
2 votes
0 answers
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Covolume of $PSL(2,\mathbb{Z})$ in $PSL(2,\mathbb{Q}_p)$

Ihara proved that every torsion-free discrete subgroup in $SL(2,\mathbb{Q}_p)$ is free and of finite covolume. $SL(2,\mathbb{Z})$ is not torsion-free. Is it a discrete subgroup and what is the ...
1 vote
0 answers
73 views

Discrete commutative subgroups of Euclidean isometries form lattice

I am trying to prove the following: If $g$ is a flat Riemannian metric on $\mathbb T^n$ (the $n$-dimensional torus), then $(\mathbb T^n, g)$ is isometric to a Riemannian quotient of the form $\mathbb ...
5 votes
1 answer
1k views

Weight lattice modulo Root lattice example

Given a compact, connected, semisimple Lie group $G$ it is known that: \begin{equation} Z(G)=\Lambda_{weight}/\Lambda_{root} \end{equation} In this question there is an explanation of why this is true....
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1 answer
142 views

Intersection of lattice and closed subgroup

Let $G$ be a locally compact group, $\Gamma$ a (uniform) lattice in $G$. Let $H \leq G$ be a closed subgroup of $G$. Is $H \cap \Gamma$ a (uniform) lattice in $H$? In the discrete case, this is easy ...