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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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 $...
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1answer
38 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 ...
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1answer
64 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{...
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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 ...
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1answer
101 views
Triangles with sides $1,k,k^2$ [closed]
Let $\Delta_k$ be the Euclidean triangle with sides of length $1,k,k^2$, where $k$ is a positive real number so that $1,k,k^2$ satisfy the triangular ineguality.
For instance, an easy example is when $...
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1answer
35 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 (...
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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 ...
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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 ...
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35 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 ...
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1answer
27 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 ...
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Relevant Voronoi vector and Shortest Vector
Possibly this is related to the question I asked yesterday here, but I am not very sure about that.
Let $L$ be an $n$-dimensional lattice. The Voronoi cell $V(u)$ is then defined as the set $\{x\in \...
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2answers
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Indecomposable elements in a lattice
Let $L$ be an discrete lattice in $\mathbb R^n$. We say that a nonzero $a\in L$ is indecomposable if and only if $a$ cannot be written as $a=b+c$ with $b,c$ nonzero and $b^T c>0$.
I was initially ...
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Arithmetic Lattices in Vector Spaces
In the literature one often sees references to discrete subgroups of semi-simple algebraic and Lie groups which are arithmetic lattices.
One can also define a lattice in a real vector space , is it ...
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Formalizing Siegel lattice integral formula
Siegel's lattice integral formula says that for $f \in L^1 (\mathbb R^n-[{0}])$, $$\int_{\mathbb R^n} f(x) dx = \int_{L_n} f^{*}(\lambda)$$ where on the right we integrate over the covolume lattices ...
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1answer
56 views
Cocompact Discrete Subgroups
I am reading a notes on homogeneous dynamics and I encountered the following statement
"any discrete subgroup $\Gamma\subset G$ such that $G/\Gamma$ is compact is a lattice of $G$". How to prove this ...
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1answer
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Finite subgroups of $SL_2({\mathbb R})\times SL_2({{\mathbb R}})$
Is there a classification of finite subgroups of $SL_2({\mathbb R})\times SL_2({{\mathbb R}})$?
For instance we have all cyclic groups and all direct products of two cyclic groups. Are there any ...
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85 views
Is every surface locally symmetric?
Can every compact connected 2 manifold be expressed as
$$
\Gamma\backslash G/H
$$
where $ G $ is a Lie group, $H $ a subgroup of $ G $, $G/H $ a symmetric space, and $ \Gamma $ a discrete subgroup of ...
2
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1answer
50 views
Ergodic action on quotient
I am reading the book "Introduction to arithmetic groups" by Dave Witte-Morris, and I am in Chpater 14 on ergodic theory. In particular I am stuck on exercise 14.2#16 which reads;
Assume $\Gamma$ is ...
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1answer
70 views
Is the map $G\to G/\Gamma$ 'locally measure preserving?'
Let $G$ be a Lie group with Haar measure $\mu$ and $\Gamma$ be a discrete subgroup of $G$. Assume that $X:=G/\Gamma$ admits a nonzero $G$-invariant Radon measure $\nu$.
Recall that the map $p:G\to G/\...
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1answer
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Cusps in fundamental domains for lattices in $SL_2(\mathbb{R})$
Let $\Gamma\leq SL_2({\mathbb{R}})$ be a Fuchsian group of the first kind (so a lattice). I understand that $\Gamma \backslash SL_2(\mathbb{R})$ is compact $\iff$ $\Gamma$ contains no parabolic ...
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2answers
76 views
Trouble with definition of a lattice: Meaning of finite volume of $\Gamma\setminus G$
Let $G$ be a Lie group and $\Gamma$ be a discrete subgroup of $G$. Definition 1.3.5 of Dave Witte Morris's Introduction to Arithmetic Groups says the following:
We say that $\Gamma$ is a lattice in ...
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1answer
138 views
convergence of vectors in $3$-dimensional sub-lattice of $\mathbb{R}^3$
Given a sequence of unimodular lattices $(\Lambda_n)_n$ in $\mathbb{R}^3$ $\big($i.e. a $3$-dimensional $\mathbb{Z}$-submodule of $\mathbb{R}^3$ with covolume $1$, meaning for any $\mathbb{Z}$-basis $...
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1answer
71 views
Is $SL_2(\mathbb{Z}[\sqrt{d}])$ a lattice in $SL_2(\mathbb{R}) \times SL_2(\mathbb{R})$ for all d?
Let $d > 0$ be a square-free integer.
If $d \equiv 2,3 \text{ mod } 4$, then $\mathbb{Z}[\sqrt{d}]$ is the ring of integers of the number field $\mathbb{Q}(\sqrt{d})$. We can thus consider the ...
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1answer
104 views
Presentations of discrete subgroups of $\textrm{PGL}_2(\mathbb{R})$
It is well known that geometrically finite Fuchsian groups, or finitely generated discrete subgroups of $\textrm{PSL}_2(\mathbb{R})$ can be classified up to isomorphism by their signature $[g,s;m_1,\...
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Dense projections of lattices in $G\times\textrm{Aut}(T)$
Let $G$ be a simple connected Lie group and let $T$ be a $k$-regular tree. Let $\Gamma$ be a lattice in $G\times\textrm{Aut}(T)$.
Assume that the intersection of $\Gamma$ with each direct factor is ...
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Does $N_{\textit{SL}_2(\mathbb{C})}(\Gamma)/\Gamma$ have an infinite number of components.
Let $\Gamma$ be a discrete co-compact subgroup of $\textit{SL}_2(\mathbb{C})$ and let $\pi:\mathcal{X}\mapsto B$ be a deformation of the homogeneous space $\textit{SL}_2(\mathbb{C})/\Gamma$. When I ...
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1answer
87 views
Irreducible lattices in $G=G_1\times G_2$
First, we shall recall the definition of an irreducible lattice.
Let $G$ be a Lie group which admits a direct product decomposition into simple non-compact factors $G_1\times\dots\times G_k$. A ...
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1answer
135 views
Lattices are not solvable in non-compact semisimple Lie groups
I'm trying to prove the following result.
If $G$ is a non compact semisimple Lie group (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is not solvable, and $[\...
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165 views
Intuition behind the definition of an irreducible lattice
I am reading Dave Witte-Morris' book on arithmetic groups and having trouble getting a handle on his definition of an irreducible lattice.
A lattice $\Gamma\subset G$ in a semisimple Lie group $G$ is ...
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1answer
65 views
Example of non-cocompact lattice in a specific topological group
An exercise in Dave Witte Morris' Introduction to Arithmetic Groups asks the reader to suppose the following.
$\Gamma$ is a non-cocompact lattice in a topological group $H$
$H$ has a compact, open ...
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1answer
153 views
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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1answer
47 views
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
66 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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0answers
100 views
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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1answer
39 views
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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1answer
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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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1answer
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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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1answer
302 views
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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1answer
562 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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95 views
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 $\...
4
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0answers
183 views
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 ...
3
votes
1answer
821 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 ...
4
votes
2answers
86 views
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 ...
1
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0answers
34 views
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 $...
1
vote
1answer
104 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 \...
