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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Discrete subgroup of complex orthogonal group
Is there any reference for the discrete subgroup of complex orthogonal group SO(n,C)? Any classification or examples?
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Does the homogeneous spaces $K^{\mathbb{C}}/{{Z(k)}^{\mathbb{C}}}$ have a natural Kähler or sympletic structure?
Let $K$ be a connected compact Lie group, $K^{\mathbb{C}}$ be complexified Lie group of $K$.
Denote $Z(k)$ by the centralizer of k∈K and $Z^{\mathbb{C}}(k) $ be the complexified Lie group of $Z(k)$ ...
- 11
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symmetric points on symmetric spaces
Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on $1,2,\...
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Estimate for the Iwasawa decomposition in loop groups
Let $GL(n,\mathbb{C})$ be the general linear group and let $U(n)$ be the unitary group in it, which is a maximal compact subgroup.
I consider the loop group $\Lambda GL(n,\mathbb{C})$ of maps from $S^...
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How can one find generators of basic differential forms on homogeneous spaces?
Dear all,
In short, my problem is that I would like to have a better control of the 1-forms on a homogeneous space. Contrary to the group case, the module of differential form is not trivialisable. ...
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algebraic closure of Lie groups in
Let $G$ be a connected, simply connected, solvable, complex Lie group with a discrete subgroup $\Gamma$.
Let also $G_a$ be Hochshild-Mostow hull of $G$, i.e., there exists a solvable linear algebraic ...
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infranilmanifolds: harmonic forms parallel?
I am studying Lott's paper : "On the spectrum of a finite volume negatively-curved manifold" and the satement is following:
We have an compact infranilmanifold $N$ which is finitely covered by a ...
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Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?
Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...
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Extensions of Groups
I believe there is a reasonable notion of $\text{Ext}^1(G,H)$ in the category of groups (where $G$ and $H$ are groups). Is there a decent reference describing this?
My particular situation involves a ...
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Representations of $\mathfrak{so}(3)$ ($\mathfrak{so}(2,1)$) and $SO(3)$ ($SO(2,1)$)
(Apologies if this question is too basic!) I have explicit 5-dimensional real representations of $\mathfrak{so}(3)$ and $\mathfrak{so}(2,1)$, and I want to know whether it's necessarily true that the ...
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$TSU(n)$ completely integrable with 3 $SU(2)$ invariant functions?
Consider the Lie group $SU(n)$ endowed with the standard bi-invariant metric. Then $SU(n)$ can be viewed as a symmetric space of $K_{n,n} := SU(n) \times SU(n)$.
Define $M := K_{n,n} /SU(n)$. Using ...
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Are lattices in the special real linear group subgroup seperable?
Let $G \leq SL_2(\mathbb{R})$ be a lattice, let $H \leq G$ be a finitely generated subgroup of infinite index, and let $n \in \mathbb{N}$. Must there be some $H \leq U \leq G$ such that $n \leq [G : U]...
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How Can I Tell when A Subgroup of a Lie Group is Generated by Unipotents?
I'm trying to understand the proof of the Oppenheim conjecture using Ratner's theorem, and I don't immediately see why $SO(2,1)$ is generated by unipotents. Why is $SO(2,1)$ generated by unipotents? ...
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Can there exist two non-equivalent equivariant actions of a group on vector bundle?
Can there exist two non-equivalent equivariant actions of a group $G$ on vector bundle over a $G$ space?
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Cartan involutions of su(n)
I have a question regarding Cartan involutions of su(n). Some sources say that there is only one up to equivalence (Wikipedia on Cartan Decomposition). Others say there are Types I, II, III. I looked ...
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Explanation of $y = x \exp(\triangle)$ for a Lie Group
Let $M$ be a non-compact matrix Lie group and $T_e M$ its lie algebra.
Consider a point $x \in M $ and $ \triangle \in T_e M$.
To move from $x$ to a point $y \in M$ along $\triangle$, below group ...
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Finding spherical representations of $GL(n, \mathbb{C})$.
I am looking for literature that might contain the spherical representations of $GL(n, \mathbb{C})$. Here a spherical representation is an irreducible representation $\rho$ of $G$ on $\mathbb{C}$ such ...
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A Criterion for Reductivity of Lie Subgroups
Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one ...
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"Homogeneity" of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]
My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
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Which groups admit a unique Lie group structure?
This question is a follow-up on the answer given here Can a Lie group as an abstract group be given more than one topology making it a Lie group?
It is motivated by the following observations:
If $m,...
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The lie algebra of the orthogonal group of an arbitrary space time metric
Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...
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equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution
I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
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The action of graph automorphism of finite symplectic group on maximal subgroups
Let $G=Sp(4,2^f)$ with $f>1$. Based on the facts when $f$ is small, I would feel the following:
$G$ has two conjugacy classes of subgroups isomorphic to $SO^+(4,2^f)$. One is in Aschbacher's class ...
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Zariski closure of orbits of real groups on complex flag manifolds
Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...
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The orthogonal group of a riemannian metric
Let the inner product of the vectors X and Y on a given four dimensional manifold (EDIT: make this R4) be defined as (X*Y) = gikXiYk; using the summation convention for repeated indicies.
Let A be a ...
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Name for a class of parabolic subgroups
This is a reference request for a (the) name of the following class of parabolic subgroups of a complex simple Lie group $G$:
Recall that parabolic subgroups of $G$, containing fixed Borel subgroup, ...
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Jordan decomposition in a classical group
Let $\mathfrak{g} \subset \mathfrak{gl}_n$ be one of the classical real or complex semisimple Lie algebras. If $g \in \mathfrak{g}$, then $g$ has a Jordan decomposition $g = g_s + g_n$ with $g_s$ ...
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reference for the slice theorem for Banach Lie group actions on Banach manifolds
I am looking for a reference treating the slice theorem for Banach Lie group actions on Banach manifolds, i.e. proving that a smooth, free and proper action of a Banach Lie group $G$ on a Banach ...
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Raising and lowering operators for SL(n,K) on homogenous polynomials
The short version:
Can the theory of weights for SL(n,C) be explained concretely in terms of raising and lowering operators on spaces of polynomials?
A deleted question asked how to prove SL(3,C) ...
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characters on unipotent group
Let $G=GL_{n}$ and $N$ the maximal unipotent subgroup, $\mathbb{A}$ the ring of adeles on a number field $F$.
We fix a non trivial character $\psi:F\backslash\mathbb{A}\rightarrow \mathbb{C}^{*}$.
We ...
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On continuous part of the L^2 spectrum
Suppose $G$ is a real reductive Lie group and $\Gamma$ is a lattice in $G$ (of finite co-volume). I am reading Langlands's paper " On the functional equation satisfied by the Eisenstein series". I ...
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How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?
By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over ...
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homomorphisms from one Lie group to another
I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie"
http://www.ams.org/mathscinet-getitem?mr=0379749
and I was wondering if I could ask for help with understanding one ...
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$v_1$-periodic homotopy and principal bundle classification
This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact ...
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Parabolic-type subgroups of GL(V)
Dear all,
Consider a flag $V=V_1\supset V_2\supset \cdots \supset V_k\supset V_{k+1}=\{0\}$ of a vector space $V$ over a field of $p$ elements. Let $I$ be a subset of the index set $\{1,2,...,k\}$.
(...
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Weil's theorem about maps from a discrete group to a Lie group.
Let K be a group (with discrete topology), G be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps ...
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Nilpotent subgroups of uniform finite index
Let $G$ be a Lie group, $K\subseteq G$ be a compact group and $N\subseteq$ be a nilpotent group s.t. $N\cap K= \{e\}$. Let $H=N\rtimes K$ be the semidirect product of $N$ and $K$ and let $\Gamma$ be a ...
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subgroups of higher rank lattices
This is related to the question $G=\langle a\rangle H$ for subgroup $H$ raised a few days ago. Suppose $\Gamma $ is a higher rank lattice (for example, $SL_3({\mathbb Z})$). As Misha says in his ...
- 11.2k
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tensor product of two irreducibles having same maximal weight
Is there any explicit decomposition of tensor product of two finite dimensional irreducible modules of simple lie algebras whose highest weights are same?
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Have derivatives of determinants along 1-psg's ever been 'coherently' computed via Jacobi's formula?
Suppose $\mathfrak{p}$ denotes all the symmetric matrices in $\mathfrak{sl}_{2n} \mathbb{R}$.
Then for each parameterized 1-dimensional linear subspace $\xi=\xi(t)$ of $\mathfrak{p}$ we get a 1-...
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Convexity radius of a Lie Group
Is there a nice formula/method to find the convexity radius of a matrix Lie group (the manifold can be noncompact) ?
Edited based on comments:
Definition : Convexity Radius (Berger - Panoramic View ...
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Complex Finite Dimensional Representation of GL(N,C)
What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$?
We already know all the complex finite dimensional linear representation of SU(N).
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Attainability of Global Optima In Optimal Control
Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...
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Groups of Hodge type, hodge structure on Lie algebra
Hi,
Let $W$ be a real algebraic group, and $G$ the associated complex group. Then $W$ is of Hodge type if there is a $\mathbb{C}^*$ action on $G$ such that $U(1)$ preserves $W$ and the action of $-1$ ...
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Parallel translation in Lie groups
Let G be a Lie group with a left invariant metric. If X and Y are left invariant vector fields and [X,Y]=0, then it is easy to show that Y is parallel to exp(tX).
But if [X,Y] is not zero, what is ...
- 454
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question about twisted group of Lie type A_n
Let $G=PSU_3(q)$ and $q=p^n$, where $n$ is odd. Can we conclude that $PSU_3(p)$ is a subgroup of $G$?
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Rationality of intersection of algebraic groups
Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and $H(\...
user70017
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Largest dimensional Lie subgroup of $SU(N)$ [duplicate]
What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension.
I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...
- 2,099
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L^2 basis of class functions on a compact Lie group that are point-wise small
Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta_1, \ldots, \theta_k) \mapsto e^{i \sum_j c_j \theta_j}$ for ...
- 4,466
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A specific question regarding a proof in Knapp's book
I got stuck in an apparently trivial point within the proof of Lemma 3.13 on p. 55 of Knapp's Representation Theory of Semisimple Groups. The author concludes in the first paragraph that $f_v$ must be ...
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