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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Is $SL(n,\mathbb{R})/SL(n, \mathbb{Z})$ a Hausdorff space?
The special linear group $SL(n, \mathbb{R})$ of degree $n$ over $\mathbb{R}$ is the set of $n \times n$ matrices with determinant $1$, with the group operations of ordinary matrix multiplication and ...
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Transcendence in $\mathbb{R}^2$
Let $\boldsymbol{v}_1=(x_1,y_1),\boldsymbol{v}_2=(x_2,y_2)\in\mathbb{R}^2$ be linearly independent over $\mathbb{R}$.
Define $\boldsymbol{v}\in\mathbb{R}^2$ to be transcendental if at least one of ...
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1answer
35 views
Are commensurable subgroups contained in a finite overgroup?
Let $\Gamma$ and $\Lambda$ be commensurable discrete lattices in a semi-simple Lie group $G$, i.e. the intersection $\Gamma\cap\Lambda$ has finite index in both $\Gamma$ and $\Lambda$. Let $\Omega=\...
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Are these enough conditions for the subgroup to be a full latice?
I was wondering how to prove the following, or if you like, whether it is true, although I am almost certain it is. Let $V$ be a finite dimensional vector space over $\mathbb{C}$, say of dimension $n$....
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Siegel Transforms on Homogeneous Spaces?
For any integer $n \geq 2,$ we may identify the space of unimodular lattices in $\mathbb{R}^n$ with the homogeneous space $X_n := \mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})$ via the ...
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Fuchsian groups in $\text{SL}(2,\mathbb{R})$ and commensurability in $\text{GL}(2,\mathbb{R})$
Let $\Gamma_1,\Gamma_2 \subset \text{SL}(2,\mathbb{R})$ be two Fuchsian groups. Assume that they are commensurable as subgroups of $\text{GL}(2,\mathbb{R})$, that is, there exists $g \in \text{GL}2,\...
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53 views
The lattice generated by $\{w(\rho) - \rho\,\vert\,w\in W\}$
Consider an irreducible root system associated to a complex simple Lie algebra $\mathfrak{g}$. Let $\rho$ be the half sum of positive roots and let $W$ be the Weyl group. Then what is the lattice $L$ ...
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23 views
Discrete subgroup of a Lie group is generated on $\mathbb{Z}$ by vectors in the Lie algebra
Let $G$ a Lie group of dimension $n$ and let $\text{exp}: LG \rightarrow G$ the exponential map. We assume that $\exp$ is a group homomorphism. We note $K:= \text{ker}(\exp)$. Since $\exp$ is a local ...
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A question about a mapping between spaces of continuous functions in LCGs
In Raghunathan's book Discrete subgroups of Lie groups we have in the chapter Generalities on Lattices the following setting: $G$ is a Locally compact group, $H$ a closed subgroup of $G$, and $G,H$ ...
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What is weight lattice modulo coroot lattice?
In Lectures on Tensor Categories and Modular Functors by Bakalov and Kirillov, the $S$ matrix (expression 3.3.7) is expressed in the form $\vert P/kQ^\vee \vert^{-1/2}\times(\cdots)$, where $k\in \...
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Margulis' Arithmeticity theorem and arithmetic manifolds / locally symmetric spaces
I admit that there are several different definitions of an arithmetic manifold in the literature, but consider the following one:
An arithmetic manifold is a quotient $M = \Gamma \backslash \mathbf{...
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Is a conjugate of a fundamental domain a fundamental domain again?
Let $G$ be a locally compact hausdorff topological group with Haar measure $\mu$, $\Gamma$ be a lattice in $G$ and $F$ be a fundamental domain of $\Gamma$ in $G$, i.e.,
(i) $F$ is Borel set with $\mu(...
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42 views
The algebraic structure of $SU(2)$
Is there any references or study for the structure of the Lie group $SU(2)$? (Such as matrixes representation)
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248 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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Discrete subgroup $\Gamma \subseteq G$ is a lattice iff the locally symmetric space $\Gamma\backslash G/K$ has finite volume
I'm trying to see how lattices can be defined in terms of the volume of the associated locally symmetric space. In Exercise §1C#6 of Dave Witte Morris book, one is asked to prove the following:
Let ...
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Computing the genus of a manifold?
I was reading week 193 of John Baez' blog, and he mentions that $E_8$ can be realized as the symmetry group of a certain 57-dimensional manifold;
Recently, some mathematical physicists have been ...
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Non-vanishing of left- vs. right-averages over lattices in $SL(2,\mathbb{R})$
EDIT: I have now asked the same question on MO.
Background. Let $G=SL(2,\mathbb{R})$, let $K=SO(2)$, and let $\Gamma$ be a lattice in $G$, e.g. $SL(2,\mathbb{Z})$. Let $\phi \in L^1(G)$ be $\...
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More general lattices on the complex plane
I have learned that lattices defined on the complex plane can be defined by $\mathbb Z\tau_1\oplus\mathbb Z\tau_2$, where $\tau_1,\tau_2\in\mathbb C$ are linearly independent over $\mathbb R$. I have ...
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1answer
366 views
Difference between the volume/covolume of a lattice
I am trying to learn some basic knowledge on lattices for studying Minkowski's theorems and Dirichlet's unit theorem. My problem is, I could not build the basics of lattice theory and there are some ...
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2answers
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Finding the volume of $\mathcal{O}$
Let $K$ be a number field, $r_1$ denotes the number of real embeddings and $2r_2$ denotes so that $r_1+2r_2 = n = [K:\mathbb{Q}]$. Define the ring homomorphism, canonical imbedding of $K$,
$\sigma:K ...
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Is there any canonical isomorphism between $L^2(L)$ and $L^2(G/L)$?
Let $L$ be a lattice in a Lie group $G$. If $L = \mathbb{Z}^n$ and $G = \mathbb{R}^n$ then $L / G = (S^1)^n$, and there is a canonical isomorphism (Fourier transform) between $L^2(\mathbb{Z}^n)$ and $...
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1answer
90 views
Limit in the quotient space of a topological group
Let $G$ be a topological group (or Lie group), $\Gamma <G$ a lattice, $H<G$ a closed connected subgroup such that $H\cap \Gamma$ is a lattice in $H$. let $K=N_G(H)\cap\Gamma$.
Let $x_i\in \...
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1answer
72 views
Do uniform lattices in simple lie groups contain free abelian subgroups?
Let $\Gamma$ be a cocompact lattice in a simple Lie group $G$ such that $G$ has rank at least $2$. Does $\Gamma$ contain a rank $2$ free abelian group?
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390 views
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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1answer
831 views
Proving that a circle will contain n lattice points?
A lattice point is a point $(x, y)$ in the plane, both of whose coordinates are integers.It is easy to see that every lattice point can be surrounded by a small circle which excludes all other lattice ...
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votes
1answer
96 views
Understanding the Action of a (Quasi-)Fuchsian Group on $\mathbb H^3$
Let $\mathbb H^3$ be hyperbolic $3$-space and let $\Gamma \subset PSL(2,\mathbb R) \subset PSL(2,\mathbb C) \cong Isom
^+(\mathbb H^3)$ be the (Fuchsian) fundamental group of a Riemann Surface $\Sigma$...
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1answer
312 views
Example of uniform tree lattice?
I'm reading about uniform tree lattices. Let $X$ be a locally finite tree and $G$ its automorphism group. A subgroup $\Gamma\leq G$ that is discrete and such that $\Gamma\setminus X$ is finite is ...
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Understanding Wikipedia's definition for lattice
I'm reading some texts about lattices in topological groups, and Wikipedia has a discussion under "Definition" in the link https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup).
I understand the ...
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1answer
61 views
Clarification in Covering Radius Problem
I am having difficulty understanding this particular definition of Covering Radius Problem - "Given a basis for the lattice, the algorithm must find the largest distance (or in some versions, its ...
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1answer
292 views
Difference between a $G$-invariant measure on $G/H$ and a Haar measure on $G/H$
Let $G$ be a locally-compact topological group, and $H$ be a normal subgroup.
$G/H$ is a locally-compact topological group as well, and if we assume $H$ to be closed then $G/H$ is Hausdorff and ...
4
votes
1answer
935 views
Relations between center (fundamental group) and (co)root and weight lattices for Lie groups
I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
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votes
1answer
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Examples of Lattices in $\operatorname{Isom}(H^n)$ for all $n \geq 2$?
I just had an exam today where I was asked to give an example of a lattice in $\operatorname{Isom}(H^n)$ for all $n \geq 2$, and with bonus points if I could give cocompact and noncocompact examples.
...
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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 $SL_n(\...
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votes
1answer
101 views
I am looking for an introduction to hyperbolic surfaces as a quotient of the upper half plane by lattices.
I keep coming across results of the form: If we take the quotient of the upper half plane by a Fuchsian group with this property, we get a surface with that property (cusps, funnels, in/finite area, .....
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1answer
62 views
What does the topology on $SL(n, \mathbb{R}) / SL(n, \mathbb{Z})$ intuitively look like?
I have come across Mahler's compactness criterion, and am having trouble wrapping my head around the topology of the moduli space of unit volume lattices.
Is there an intuitive way to think about it, ...
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votes
2answers
275 views
Easy examples of non-arithmetic lattices
I'm starting to look a bit more at discrete subgroups of Lie groups, particularly lattices. A lot is written about arithmetic lattices of Lie groups, and examples abound.
It appears that much less is ...
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2answers
965 views
What is the Haar measures on $SL(2, \mathbb R)$ And $SL(2,\mathbb R) / SL(2,\mathbb Z)$?
How does one parametrize those spaces in order to do integration over them?
What's a good reference for doing integral a with Haar measures over matrix groups?
3
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1answer
37 views
Scalar multiple of one lattice contained in another
I believe my question boils down to the following question:
Given lattices $L$ and $L'$ in $k^{n}$, does there exist $\lambda \in k^{\times}$ so that $\lambda L' \subseteq L$ and $\lambda L' \not\...
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115 views
What is a Complete Set of Weights of a Representation of a Lie Subalgebra?
In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations.
I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
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2answers
377 views
What is the dual lattice of Kagome lattice?
We know that the dual lattice of a triangular lattice is the honeycomb lattice. What is the dual lattice of Kagome lattice?
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1answer
118 views
Weyl group and weight lattice chambers.
Consider two simple Lie groups $G_1$ and $G_2$. Let $G_1$ have $W_1$ as a Weyl group and $G_2$ have $W_2$ as a Weyl group.
Is it true that the Weyl group of $G_1 \times G_2$ is $W_1 \times W_2$?
...
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1answer
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Why has $\mathrm{GL}(n, \mathbb{R})/\mathrm{GL}(n, \mathbb{Z})$ infinite co-volume?
The space $X=\mathrm{SL}(n, \mathbb{R})/\mathrm{SL}(n, \mathbb{Z})$ can be identified as the space of unimodular lattices in $\mathbb{R}^n$ and it is well-known that if we take the Haar measure on $\...
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1answer
37 views
Questions about definition of edges in affine building.
I have a question about about edges in a building in the book of expander graphs by Alexander Lubotzky, page 69.
We know that if $L_1' \subseteq L_2'$ and $[L_2' : L_1'] = p$, then there is an edge ...
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0answers
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Classification of 6-D Nilmanifolds
I am reading the G.Cavalcanti and M.Gualtieri's Generalized Complex Structures on Nilmanifolds. In the introduction it is said that there are 34 nilpotent lie algebra isomorphism classes. There are ...
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1answer
247 views
Example of a discrete uniform subgroup of a Lie group which is not virtually torsion-free
I'm looking for an example, or source of examples, of a discrete uniform subgroup $G$ of a Lie group $\Gamma$, with $G$ not virtually torsion-free. By uniform subgroup I mean that the quotient $\...
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35 views
Enlightening explanation of a theorem of Zimmert's
I'd like to know wether anyone has ever read an enlightening explanation (e.g. with geometric argument) of the following paper:
Zimmert, R.
Zur $SL_2$ der ganzen Zahlen eines imaginär-quadratischen ...
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1answer
247 views
Infinite index subgroups of $SL(3,\mathbf{Z})$
Finite index subgroups of $SL(3,\mathbf{Z})$ are well-know (at least we know that they are congruence subgroups).
But I wasn't able to find reference on infinite index subgroups.
Does someone knows ...
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0answers
76 views
“p-adic” presentation of surfaces
On several occasions I heard about the following result:
For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
3
votes
1answer
210 views
Lattices inside matrix groups $SL_2(K)$
I am currently a second year undergraduate majoring in math and our university is offering an opportunity for undergraduates to do a project over the summer break. I have spoken to my professor who is ...
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1answer
280 views
Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$
What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$?
$\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
