Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
2,938
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Centralizers of one parameter subgroups in semi-simple Lie groups
Suppose G is a connected semi-simple Lie group with finite center, and A, B are one parameter subgroups of the same Cartan subgroup. If the connected components of the identity of the centralizers of ...
6
votes
3
answers
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Good book on representation theory of GL(n)
I am interested in a recommendation for a good book which discuses representation theory of GL(n)(say over field of complex numbers).
I know only a basic representation theory.
The question I am ...
6
votes
1
answer
877
views
Uniform lattices in semisimple Lie groups
Let $\Gamma$ be a uniform lattice in a semisimple Lie group $G$.
Must $\Gamma$ be virtually torsion-free?
If (1) is true, then does this work more generally if $G$ is reductive?
I am motivated by a ...
6
votes
2
answers
1k
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Lattices in SOL
Consider a semi-direct product $\mathbb{Z}^2\rtimes_A\mathbb{Z}$, where $A\in SL_2(\mathbb{Z})$ and $|Tr(A)|>2$. It is clear that it is isomorphic to a lattice in the 3-dimensional solvable Lie ...
6
votes
1
answer
412
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Does $\pi_1(H)=0\Rightarrow \pi_3(G/H)=0$ for a simple and simply connected Lie group $G$?
$\DeclareMathOperator\SU{SU}$Let $G$ be a simple and simply-connected Lie group and $H\neq 1$ be a simple and simply connected subgroup, is it true that $\pi_3(G/H)=0$? If not, what is a counter-...
6
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1
answer
370
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Do the exceptional root systems arise in the real world?
I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...
6
votes
1
answer
492
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Computing a Commutator Subgroup
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I’m studying the group $\O(5,5,\mathbb{Z})$, the indefinite orthogonal matrices with integer entries. In particular, I ...
6
votes
1
answer
531
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Finite simple groups and $ \operatorname{SU}_n $
A follow-up question to Alternating subgroups of $\mathrm{SU}_n $.
$\DeclareMathOperator\PU{PU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PU_m $ be the projective unitary group, ...
6
votes
1
answer
427
views
Hyperbolic manifolds with infinite cyclic fundamental group
It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act ...
6
votes
2
answers
297
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Irreducibility of Gelfand-Serganova strata
To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $...
6
votes
2
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481
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The Analog of Borel Subgroup in a Compact Real Form
I recently learned that there is a one-to-one correspondence between isomorphism classes of complex reductive groups and isomorphism classes of compact connected real Lie groups given by taking a ...
6
votes
1
answer
268
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Simultaneous triangularisation of an exterior power of a set of matrices
I'm working on some research problems relating to random matrix products, and this is taking me into areas of mathematics I've not previously studied: Lie groups, representation theory, and real ...
6
votes
1
answer
2k
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Difference between the Laplacian and the sub-Laplacian of a Lie group
Given a Lie group $G$, what is the difference between the Laplacian $\Delta$ and the sub-Laplacian $\Delta_{sub}$ of $G$. And what are the properties that we lose when going from sub-Laplace to ...
6
votes
2
answers
691
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Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?
Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is $\text{Lie}(\...
6
votes
1
answer
430
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geodesic of $\rm SO(3)$ as a compact Lie group vs as a Riemannian symmetric space
I got a little bit confused about the definition of geodesic for $\rm SO(3)$ as
a compact Lie group
a Riemannian symmetric space
In the former case, it is given by the usual matrix exponential:
$$
\...
6
votes
1
answer
814
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Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?
I'm asking a question about Lie group representation.
Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...
6
votes
1
answer
252
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Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky
The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, \...
6
votes
2
answers
524
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Do geodesics in SL2R map to geodesics in the hyperbolic plane?
I am looking for a reference/proof/disproof of the following statement.
Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...
6
votes
1
answer
489
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Root space decomposition
What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form
$\left(
\begin{array}{cc}
X & Y \\
\overline{Y}^t & Z \\...
6
votes
1
answer
536
views
Stiefel manifolds and polar decompositions
The real Stiefel manifold $V_{n,k}$ of orthogonal $k$-frames in $\mathbb{R}^n$ can be viewed as the reductive homogeneous space $G/H=O(n)/O(n-k)$. If ${\frak{so}}(n)$ is the Lie algebra of $O(n)$, ...
6
votes
1
answer
1k
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Decomposition of semisimple Lie group into almost simple factors
Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk ...
6
votes
2
answers
792
views
Classification of real forms up to inner automorphisms
I hope to know the classification of real forms of complex simple Lie algebras of types $A$, $D$, $E$ up to inner automorphisms.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be real forms of a complex ...
6
votes
1
answer
168
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Does every Lie algebra appear as centralizer of an element in a semisimple Lie algebra?
Given a finite dimensional, complex, semisimple (fcss) Lie algebra $\mathfrak{g}$ and an element $x\in\mathfrak{g}$, denote by $\mathfrak{g}^x$ the centralizer of $x$ in $\mathfrak{g}$ i.e. the set $\{...
6
votes
1
answer
226
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Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?
Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...
6
votes
1
answer
304
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Number of points on a linear algebraic group over a finite field
Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be a good, simple lower bound for $G(F_q)$?
One can get something fairly nice ...
6
votes
1
answer
301
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How to distinguish conjugacy classes in SO(2n) efficiently?
In a compact connected Lie group $G$, each element is conjugate to an element of a maximal torus $T$. For a classical group, one can pick a basis of the tautological representation such that $T$ is ...
6
votes
1
answer
530
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Torus bundles and compact solvmanifolds
I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m ...
6
votes
1
answer
353
views
Equivariant implicit function theorem
Let $f:\mathbb{R}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a smooth function and $G\subset \operatorname{SO}(n)$ be a $1$-dimensional compact Lie group (diffeomorphic to the circle). ...
6
votes
2
answers
372
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About Lie group $G$ has this escape property?
Every Lie group $G$ has the following escape property: For every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$.
...
6
votes
1
answer
200
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Fixed space of maximal torus and Weyl group
Let $G$ be a compact connected Lie group and $T\subset G$ a maximal torus. Let $V$ be a representation of $G$ and $U=\{v\in V: tv=v\textrm{ for all }t\in T\}$. For any $g\in N(T)$ we have for all $t\...
6
votes
1
answer
416
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Analog of the Lie Product formula for commutators
Let $X, Y$ be elements of a Lie algebra. Consider the group $G$ generated by (limits of) arbitrary products of the elements
$$ G = \langle{e^{tX},e^{sY}\rangle}$$
for all $t,s$. The Lie product ...
6
votes
1
answer
290
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Idempotent functions on Sp(1)
The quaternion group $Sp(1)\simeq S^3$ can be understood as $(z,w)\in\mathbb {C}^2$ with $|z|^2+|w|^2=1$ where multiplication is defined by $(z,w)(t,s)=(zt-\bar{s}w,zs+\bar{t}w)$.
Question: How do ...
6
votes
1
answer
1k
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Classification of compact connected abelian groups
It is known that torsion-free compact abelian groups are exactly the product of the maximal solenoid group $\Sigma_{(2,3,\cdots)}$ (which is the Pontryagin dual of the additive group $\mathbb{Q}$ of ...
6
votes
2
answers
317
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Almost free actions on simply-connected spaces
Let $G$ be a simply-connected compact topological group (you can think of $SU(n)$ if you like it more concrete), and let $X$ be a finite-dimensional simply-connected $G$-CW-complex. If we know that ...
6
votes
1
answer
541
views
Conjugacy classes of involutions in compact simple Lie group
Is there a known set $S=\{x \in G: x^2=1, x\ne1\}$ of elements of a simple compact Lie group $G$ ? By simple compact Lie group I consider $SO_n$, $SU_n$, $Sp_n$, $G_2$, $F_4$, $E_6$, $E_7$, $E_8$. (...
6
votes
1
answer
602
views
Zariski closure is semisimple
Suppose that $\Gamma$ is a finitely generated semi-group of $SL(n,\mathbb{Z})$ which acts strongly irreducible on $\mathbb{R}^n$, i.e. there is no finite union of proper nonzero linear sub-spaces of $\...
6
votes
1
answer
560
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Centreless semisimple Lie group that is not real algebraic
Let $G$ be a connected semisimple Lie group with trivial centre and $\mathfrak{g}$ its Lie algebra. The adjoint representation of $G$ defines an isomorphism of $G$ onto the connected component of the ...
6
votes
1
answer
577
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Vector fields, diffeomorphism subgroups and lie group actions
Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X_j\} \in Vect(M)$ be a ...
6
votes
2
answers
499
views
The action of the center on the extended Dynkin diagram
Let $R$ be an irreducible root system with a basis $\Pi$.
We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$.
Let $Q^\vee\subset P^\vee$ ...
6
votes
1
answer
335
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Does there exist a categorical treatment of root data(systems)?
What I am looking for is an abstract description of root data with their morphisms(!) plus a comparison with the categories of reductive groups over some field, Dynkin diagrams, Lie algebras, ...
6
votes
2
answers
705
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Measuring how far from being cocompact a lattice is
Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete
subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$
that is invariant under the action of $G$ by left-...
6
votes
2
answers
900
views
Reference Request: Steinberg's 1975 paper "On a paper of Pittie"(retrieved)
I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference ...
6
votes
1
answer
385
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locally-free Lie group action not preserving any measure
I'd like to know if there exists a connected Lie group $G$ and a closed manifold $M$ such that there is a locally-free smooth action $G\times M\to M$ (i.e. the stabilizer of any point of $M$ is a ...
6
votes
1
answer
680
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different Shimura data with common underlying group?
A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge ...
6
votes
1
answer
275
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Words in two infinitismal rotations
I asked this as subquestion in a comment pursuant to my Banach-Tarski
question. I think it is worth promoting here to a question in its own right.
Consider these two matrices over ${\Bbb R}[[\...
6
votes
2
answers
861
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Why can the Dolbeault Operators be Realised as Lie Algebra Actions
I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical case here: Recall that ...
6
votes
1
answer
297
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Is there a representation of $\mathrm{SU}_8/\{\pm 1\}$ that doesn't lift to a spin group?
$\newcommand{\GL}{\mathrm{GL}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\SU}{\mathrm{SU}}\newcommand{\Spin}{\mathrm{Spin}}\renewcommand{\O}{\mathrm
O}\newcommand{\R}{\mathbb
R}\newcommand\Z{\mathbb Z}$...
6
votes
1
answer
416
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Is every finite subgroup the integer points of a linear algebraic group?
Cross Posting this from MSE since it's been there for almost a month and it got a couple upvotes but no answers. MSE link Is every finite subgroup the integer points of a linear algebraic group?
Let $ ...
6
votes
1
answer
219
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Do weight vectors live between the highest and lowest weights?
For a simple complex Lie algebra $\frak{g}$, let $V$ be an irreducible $\frak{g}$-module. Is it true that the weights of the non-zero weight vectors in $V$ are less than the highest weight vector and ...
6
votes
1
answer
1k
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When a free action gives rise to a $G$-principal bundle
When a free action gives rise to a $G$-principal bundle
Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that
$G \backslash X$ is Hausdorff. (equivalently the image of ...