Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
3,059 questions
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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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Cocompact lattices in $\mathrm{Sp}(n, 1)$
This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...
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When is a unitary group over a ring of integers dense?
Let $ SU_n(O_d) $ denote an integral unitary group of $ n \times n $ matrices over a totally real number field $ K_d:=\mathbb{Q}(\cos(\frac{2\pi }{d})) $ where $ O_d $ is the ring of integers of $ K_d ...
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Which Lie groups are a central extension of an algebraic group?
Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an ...
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What can lattices tell us about lattices?
A general group-theoretic lattice is usually defined as something like
A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
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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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Group theory in machine learning
I'm a Machine Learning researcher who would like to research applications of group theory in ML.
There is a term "Partially Observed Groups" in machine learning theory which has been ...
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Compact Lie group has finitely many Lie primitive subgroups
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$A closed subgroup $ \Gamma $ of a Lie group $ G $ is called Lie primitive if it is not contained in any proper ...
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Existence of a bounded measurable subset of $\text{SL}(d,\mathbb R)$ that is Borel isomorphic to $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$?
$\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$ has two interesting properties: on one hand it is non-compact, but on the other hand it admits a unique $\text{SL}(d,\mathbb R)$-invariant finite ...
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Complete representation theory of $\mathrm{SL}(2,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...
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Classification of (not necessarily connected) compact Lie groups
I am looking for a classification of compact (not necessarily connected) Lie groups. Clearly, all such groups are extensions of a finite "component group" $\pi_0(G)$ by a compact connected ...
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Existence (or non existence) of principal bundle charts compatible with an $f$-reduction
I asked this question on math stack exchange here, but I wonder if it would be better received here.
Let $\pi:P\rightarrow M$ and $\pi':P'\rightarrow M$ be principal $G$ and $H$ bundles respectively, ...
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Commuting time dependent vector fields and pullback invariance
Let $X_t, Y_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields.
Is there some analogue of the following fact in finite ...
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Decomposition of tensor powers of the vector representation of $\frak{sl}_n$
Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as
$$
V(\pi_1) \...
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Frobenius norm bounds on exponentials of anti-Hermitian matrices
Suppose $X$ and $Y$ are two anti-Hermitian matrices satisifying $\|X\|, \|Y\| \leq \pi$, where $\|\cdot\|$ is the spectral norm. I'm trying to prove the following bounds on the Frobenius norm of the ...
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A compact Lie group $G$ acting on a compact Lie group $K$ transitively. Is there a $C$ such that $d(gx,gy)\leq Cd(x,y)$?
Let $G$ be a compact connected Lie group acting transitively and smoothly on another compact Lie group $K$. Let $d$ be the distance in $K$ that is not $G$-invariant. Is there a constant $C$ such that $...
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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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Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?
Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ...
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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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Online References for Cartan Geometry
I would like to learn more about Cartan Geometry ("les espaces généralisés de Cartan"). I ordered Rick Sharpe's book "Differential Geometry: Cartan's generalization...", which would take a long time ...
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On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups
I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE.
I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $...
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The map $k \mapsto \mathbf{PGL}_2(k)$
Consider the map $\zeta: \{ \mbox{division rings} \} \mapsto \{ \mbox{groups} \}: k \mapsto \mathbf{PGL}_2(k)$.
Is this map known to be an injection - in other words, if $k$ and $k'$ are nonisomorphic ...
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Injectivity of the cohomology map induced by some projection map
Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...
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Is there a general method for computing finitely generated normalizers?
I'm looking to compute normalizers of finite subgroups of $\mathrm{GL}(n, \mathbb{Z})$ and its possible that they are infinite but they are always finitely presented. For $\mathrm{GL}(n, \mathbb{Z})$ ...
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Classification of (compact) Lie groups
I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your ...
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Centralizers and algebraic groups
Suppose $G$ is a linear algebraic group - I am also happy to assume $G$ is a simple algebraic group over an algebraically closed field of characteristic zero, but the question won't require this.
The ...
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Semisimple compact Lie group topologically generated by two finite order elements
Edit: I'm specializing this question to the compact case I'll ask about the noncompact case as a new question.
Let $ G $ be a compact connected semisimple Lie group.
Do there always exist two finite ...
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What do the Carnot groups act on?
My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive.
A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...
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Image of minimal degree representation of quasisimple group unique up to conjugacy
Let $ G $ be a quasisimple finite group. Let $ d_{min} $ be the minimum dimension of a nontrivial irrep of $ G $. Must it be the case that the image of all (nontrivial) dimension $ d_{min} $ irreps of ...
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Peterson's quantum cohomology of G/P lectures
Dale Peterson famously gave a series of lectures on the quantum cohomology of flag varieties $G/P$ at MIT in 1997. These lectures are often cited in subsequent papers by other authors on the subject (...
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What is special to dimension 8?
Dimension $8$ seems special, as the partial list below might indicate.
Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$?
Surely it isn't it, in the end, simply because ...
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Is there any results about the stable (or unstable) cohomology operations on cohomology of Lie groups?
$\DeclareMathOperator\SU{SU}$For the $\mod p$ singular cohomology of classical Lie groups, such as $H^*(\SU(n); \mathbb{Z}/p\mathbb{Z})$, there are well known results about the actions of the stable ...
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Howe duality vs first fundamental theorem in invariant theory
I'm working on Howe duality, and R. Howe proved that the Howe duality of $\mathrm{GL}_n$ is equivalent to the first fundamental theorem (FFT) in invariant theory. So, Howe duality gives a ...
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Are the integer points of a simple linear algebraic group 2-generated?
Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
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Learning about Lie groups
Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
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What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
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The definition of a homogeneous vector bundle
For a homogeneous space $G/H$ a homogeneous vector bundle has a total space of the form $G \times_{\rho} V$, where $(V,\rho)$ is a representation of $H$ and $G \times_{\rho} V$ is the set of ...
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Sub-coroot systems
Let $T$ be a maximal torus of a compact Lie group $K$,
and let $\Psi \subset {\mathfrak t}$ be the (finite) set of coroots for $(K,T)$, where $\mathfrak t$ is the Lie algebra of $T$.
Assume now that $...
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Question on KAK decomposition
Let $G$ be a semisimple Lie group and let $G = KAK$ be a Cartan decomposition.
For $\mathrm{SL}_2(\mathbb{R})$ it holds for every $g \in G$ that $KgK = Kg^{-1}K$.
Does the same hold for every ...
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Algorithmically handling the Spin groups in larg(ish) dimensions
Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$? By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute ...
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Proof of Zimmer's cocycle super-rigidity theorem
I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
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What is the form of the incomplete Eisenstein series on PGL_2(C)?
Let $F$ be an imaginary quadratic number field. Let $G = \mathrm{PGL}_2(\mathbb{C}) $ and $\varGamma = \mathrm{PSL}_2(\mathcal{O}_F)$. We have the Iwasawa decomposition $G = NAK$ where $K = \mathrm{...
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Is a Lie subgroup whose center is closed, a closed subgroup itself?
I want to show that a certain Lie subgroup (i.e. generated by the exponential of elements in some Lie subalgebra) of a Lie group is closed. My knowledge of the subject of Lie groups is rudimentary, ...
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Number of representations of a semisimple Lie algebra of any given dimension
For a semisimple complex Lie algebra $\frak{g}$ it is well known that irreducible finite-dimensional representation are not characterised by their dimension.
More formally, let us define an ...
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Analogue of Margulis height function in non lattice subgroups
I have been reading this paper https://link.springer.com/article/10.1007/s11854-017-0033-4 on singular system of linear forms and non escape of mass in homogeneous spaces $G/\Gamma$ where $ G=SL(m+n,\...
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Question on Artin's Gamma function on $\operatorname{SO}(2,0)(\mathbb R)$
$\DeclareMathOperator\SO{SO}$Let $G=\SO(2,0)(\mathbb{R})$, a quasi-split group with signature $(2,0)$. Let $e$ be an element in $O(2,0)(\mathbb{R}) \setminus \SO(2,0)(\mathbb{R})$.
Let $\pi$ be an ...
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Knapp's proof that the fundamental group of a compact semisimple Lie group is finite
In Knapp's book Representation Theory of Semisimple Groups: An Overview Based on Examples, he proves the following theorem of Weyl: If $G$ is a compact connected semisimple Lie group, then $\pi_1(G)$ ...
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Quaternion representation and Haar measure of $SU(3)$ [closed]
Do we have easily and practically useful quaternion representation for $SU(3)$ group element and for Haar measure?
Also, is $SU(2)$ really simplified in the quaternion base?
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Asymptotics of Haar moments on general Lie groups
I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
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Example of a manifold which is not a homogeneous space of any Lie group
Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...
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