Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
358 questions
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What is miraculous about the mirabolic subgroup?
I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...
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3
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Matrix expression for elements of $SO(3)$
Hi all. Is there any explicit matrix expression for a general element of the special orthogonal group $SO(3)$? I have been searching texts and net both, but could not find it. Kindly provide any ...
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2
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Extension of the Hall-Witt identity
Although this question is mostly out of curiosity (as of now), I hope it is nevertheless suitable for MO.
This very recent (and still open) question about the Hall-Witt identity led me to wonder:
...
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11
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1
answer
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Alternating subgroups of $\mathrm{SU}_n $
$\DeclareMathOperator\PSU{PSU}$Let $ \PSU_n $ be the projective unitary group. Let $ A_m $ be the alternating group on $ m $ letters.
$ A_5 $ is a maximal closed subgroup of $ PSU_2 \cong SO_3(\mathbb{...
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4
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Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$.
An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...
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11
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2
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755
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Quasi-isometric rigidity of Nil
Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...
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Non-isomorphic complex Lie groups with the same exceptional Lie algebra for $\mathfrak{g_2,f_4,e_6,e_7,e_8}$?
An exceptional complex Lie algebra is a simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five such Lie algebras: $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}...
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3
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Can we promote to a Lie Group Isomorphism?
We regard an isomorphism of Lie groups to mean a group isomorphism which is simultaneously a diffeomorphism of the underlying smooth manifold. I'm wondering about how much rigidity is imposed by this ...
- 1,261
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2
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Interpret Fourier transform as limit of Fourier series
Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n =\text{Hom}(\...
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3
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Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 180
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Connected compact manifolds with unique Lie group structure
I am sorry if my question is stupid (or very hard) or common knowledge, or should be placed at math.stackexchage.com.
As long as a math student read the definition of Lie group, several natural ...
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3
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Parameterizing rotations of a cube
For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$ In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$. ...
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Algebraicity of holomorphic representations of a semisimple complex linear algebraic group
Let $G$ be a complex linear algebraic group, given to us as a closed subgroup of some $\mathrm{GL}(n,\mathbb{C})$. Suppose moreover that $G$ is semisimple. Then it's a fact that every finite-...
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what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?
Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
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Reference requested: Random walk on groups
I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...
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2
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Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups
Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to {\...
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Fake degrees: why coinvariant algebra and classical groups over finite fields?
Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated.
...
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Is endoscopy interesting in simply-laced cases?
Let $G$ be a complex algebraic group, and write $Z(g)$ for the centralizer of a semisimple element $g$ in $G$. I will assume $G$ is simply connected, in which case $Z(g)$ is connected.
Let $G^\vee$ ...
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2
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Homotopy properties of Lie groups
Let $G$ be a real connected Lie group. I am interested in its special homotopy properties, which distinguish it from other smooth manifolds
For example
$G$ is homotopy equivalent to a smooth compact ...
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4
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710
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Palais's and Kobayashi's theorems on automorphism groups of geometric structures
My question concerns two results in the neighborhood of the standard theorem of Myers-Steenrod that isometry groups of Riemannian manifolds are Lie groups. Both appear in the first chapter of ...
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2
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657
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Number of reduced decompositions of the longest element of the Weyl group
Let $R$ be a reduced root system, $W$ the associated Weyl group, and $w_0 \in W$ the longest element of $W$. In general $w_0$ admits more than one reduced decomposition into a product of reflections, ...
- 219
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A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition
$$
T(M) = A \...
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2
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715
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Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$The exceptional isomorphism $\Spin(5,1)\simeq \SL(2,\mathbb{H})$ is well-known, and I can find references that ...
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Concrete description of an exceptional minuscule variety
Let $G$ be a complete reductive Lie group. A simple root $\alpha$ is said to be minuscule if the multiplicity of the coroot $\alpha^\vee$ in $\beta^\vee$ is at most $1$ for all positive roots $\beta$. ...
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Cohomology of the unitary group
The de-Rham cohomology ring of U(n) is the exterior algebra generated by the odd-dimensional classes x_1, x_3, ..., x_(2n-1). Moreover, on a Lie group every cohomology class is represented by a unique ...
- 1,232
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5
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677
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Occurrence of semi-spin groups
In the classification of simple Lie algebras one has the familiar picture of 4 families, $A_n$, $B_n$, $C_n$ and $D_n$, and 5 exceptional groups, $F_4,$ $G_2,$ $E_6$, $E_7$ and $E_8$. The $D_n$ ...
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1
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Principal congruence subgroups in higher rank
I don't seem to have seen any explicit generators for the principal congruence subgroups of $SL(n, \mathbb{Z}),$ for $n>2,$ although it is known (Sury+Venkataramana) is that the number of ...
- 96.4k
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360
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Finding $U,V$ in Thompson's Formula
Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that:
$e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$
Given $a,b \in \mathfrak{su}(4)$ defined by:
$a=J_x ...
- 2,099
8
votes
2
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925
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Exponential map
Does there exist an infinite dimensional Lie group $G$, with Lie algebra $\mathfrak g$, such that the exponential map $exp:\mathfrak g \to G$ is not defined?
If so, can one provide an example of such ...
8
votes
1
answer
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Wedge Product of Lie Algebra Valued One-Form
I've been reading about the formal structure of gauge theories and am a little confused by the notation. Could someone clarify this for me?
Suppose that $A$ is a Lie algebra valued 1-form ...
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1
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Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds
Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
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8
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1
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A question on eigenvalues
Let $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$ be linearly independent Hermitian matrices in the the space of $6$ by $6$ Hermitian matrices as a vector space over $\mathbb{R}$. Does there always ...
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What is $SL(2,\mathbb{R})$-Chern-SImons Theory?
I found in physics that Chern-Simons theory is closely related with three dimensional gravity.
From this paper Three Dimensional Gravity Revisited, the author talks about the Chern-Simons for
$$\...
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2
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362
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Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?
Problem. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G/H$ is contractible. Is $G/H$ homeomorphic to a Euclidean space $\mathbb R^n$ for some $...
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Lie algebras to classify Lie groups
What does the classification of Complex Semi-simple Lie algebras buy us in terms of classifying Lie groups? Certainly it classifies complex semi-simple lie groups but can we get any better? I know we ...
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Is there a Levi decomposition for Lie group and algebraic group?
Let $G$ be a Lie group and $R$ be the largest connected solvable
normal subgroup of $G$.
Question 1
Is there a Lie subgroup $S$ such that: (1) $G=SR$; (2)
every real representation of $S$ is ...
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votes
1
answer
387
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What kind of locally symmetric space is a rational sphere
Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...
- 1,111
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666
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Approximating Lie groups by finite groups
How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$...
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1
answer
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The Hypercomplex Structure of $SU(3)$
(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\...
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1
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Weingarten function for unitary group
Studying integration over unitary group I came across this function, the Weingarten function Wg, such that
$$ \int_{\mathcal{U}(N)} \prod_{k=1}^{n} U_{i_kj_k}
U^*_{m_k r_k} dU=\sum_{\tau,\sigma\in S_n}...
- 1,549
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1
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673
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Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
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1
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740
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Is $Spin(N)$ a subgroup of $SU(N)$
Is $Spin(N)$ a subgroup of $SU(N)$? If so, how can we embed $Spin(N)$ into $SU(N)$? I would love to find a representation where both $Spin(N)$ and $SU(N)$ act faithfully and see explicitly how the ...
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1
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Quotienting $SU(3)$ by $U(1)$?
As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...
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1
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Harish-Chandra isomorphism for compact symmetric spaces
I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
Below ...
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1
answer
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What does the group action of a rough path in a Lie group look like?
Rough paths can be thought of as taking values in a Lie group embedded in the tensor algebra of $\Bbb R^d$. See page 17/section 2.3. Lie groups represent the continuous symmetries of some object. ...
user69208
7
votes
2
answers
412
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Homomorphisms from higher rank lattices with infinite center to $\mathbb{Z}$
Suppose that $\Gamma$ is an irreducible lattice in a semi-simple real Lie group $G$ of higher rank (with infinite center!), is every homomorphism $\Gamma \to \mathbb{Z}$ trivial?
The case where $G$ ...
- 1,101
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votes
1
answer
190
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Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds
It is known that closed spherical and hyperbolic 3-manifolds are rigid. I.e., if two such manifolds are diffeomorphic, then they are isometric (moreover, I think, that every diffeomorphism is isotopic ...
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votes
1
answer
2k
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Automorphism group of the special unitary group $SU(N)$
Let us consider the automorphism group of the special unitary group $G=SU(N)$.
We know there is an exact sequence:
$$
0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.
$$
For $G=SU(2)...
- 2,147
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votes
0
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508
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Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
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1
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Does complexified isometry group act transitively on tangent bundle of compact Riemannian manifold?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^...
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