Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
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Exponential analogue of formal connections
Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, $\...
- 2,681
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2
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Four Sphere Fibrations
Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
2
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Deligne-Simpson problem for classical groups
Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n =...
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Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations
I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
$U_iU_{i+1}U_i=...
- 71
3
votes
1
answer
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Lie Symmetries of the Bessel Differential Equation
The Bessel differential equation has an arbitrary looking form, but a lot is known about it.
$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$
Is there a way to derive the Bessel ...
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votes
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odd length Chevalley relations (in rank two)
The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: \...
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Generalized Gaussian Decomposition
Let $G$ be a connected complex semisimple Lie group. Let $H$ be a maximal torus of $G$, let $W$ be the Weyl group of $G$, and let $N_\pm$ be a pair of opposite maximal unipotent subgroups. For each ...
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Representation theory and associated bundles
I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
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Intuition for the Cartan connection and "rolling without slipping" in Cartan geometry
Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$.
The Cartan connection is supposed to formalize what it means to "roll ...
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Kirillov orbit Method for Complex nilpotent groups
Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...
- 2,468
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Samelson Products in $SO(n)$
Given a topological group $G$ one forms the commutator $c\colon G\times G\rightarrow G$, $(x,y)\mapsto xyx^{-1}y^{-1}$. This map then factors through the smash $G\wedge G$. This map is the most ...
- 5,016
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Solving equations in SO(3) : an open problem by Jan Mycielski
I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, http://www.jstor.org/stable/...
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Technical issue in the approach to Lie groups taken in a book
I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...
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1
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Totally geodesic subgroups in Lie groups
Let $G$ be a Lie group with a left invariant metric $g$.
Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every $...
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Construct discrete series of SL(2,R) as kernel of twisted Dirac operators
I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).
The idea is that each non-trivial ...
- 288
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An analogue of Deligne--Lusztig theory for real groups?
I am considering the following analogue of Deligne--Lusztig theory:
Take e.g. $G=GL_n(\mathbb{C})$, and let $F$ be the complex conjugate, then we have
$G^F=GL_n(\mathbb{R})$. Consider the ``Lang map''...
- 1,534
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votes
1
answer
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Lie functor preserves "surjections" in synthetic differential geometry?
In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism.
As pointed out ...
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votes
1
answer
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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}...
- 1,417
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votes
2
answers
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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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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}(\...
- 1,856
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1
answer
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Maximal split torus of universal chevalley group
Let $G$ be simply connected chevalley group over a field $K$. I am following the notations as in 'Lectures on Chevalley group' by Steinberg (Yale lectures). Let $H$ be the subgroup generated by $\{h_{\...
- 685
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votes
2
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514
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Can one describe the multiplication of two Bruhat cells?
For $G$ a simple linear algebraic group and $B$ a fixed Borel subgroup, we have the Bruhat decomposition $G = \coprod_{w \in W} B\dot{w}B$, where $W$ is the Weyl group and $\dot{w}$ is any ...
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votes
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The action of $GL_{\infty}$ on the infinite wedge space
This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...
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Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: $...
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When can a locally compact group be approximated by discrete subgroups?
This question is about partitioning a (locally) compact group into cells by using discrete subgroups.
Let $G$ be a locally compact group. (I am really most interested in the case where $G$ is a ...
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Are singular critical points isolated for control systems on compact semisimple Lie groups
Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form:
$\frac{d U_t}{dt} = (A + w(t)B)U_t$
where $A,B \in \mathfrak{su}(n)$ generate the ...
- 2,069
3
votes
1
answer
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Characterization of $L^1(\text{SL}(3,\mathbb{R}))$ [closed]
Is there a characterisation of the integrable functions on SL($3,\mathbb{R}$) or an explicit expression for the Haar measure?
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201
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Manifolds as simultaneous coset spaces
Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of ...
- 151
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0
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95
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G-invariant functions on manifold for G compact
In a paper I saw the following statement:
Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...
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votes
2
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319
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Commutator 2-forms on Lie groups
Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra.
For any $f$ in the dual space $\mathfrak g^*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto f([A,B])...
- 8,046
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1
answer
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Concept of Facets in the structure of reductive algebraic groups
Where can I find a precise definition of Facet ? In some online notes it is stated that Facet is a maximal subset of co-characters having the same sign for every root. But shouldn't then every facet ...
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1
answer
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$E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?
Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
- 85
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votes
1
answer
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$\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?
Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
- 85
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2
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Abelian isometry groups of codimension one
Good day.
Let (M,g) be an n-dimensional Riemannian manifold (complete, if you wish), and suppose that there exists an n-1 dimensional Abelian group acting by isometries on M. Or locally, near a point ...
- 31
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0
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Rigidity of lower-dimensional lattices in Euclidean groups
Informal intro / motivation:
Suppose I have an infinite set of atoms arranged in a 2D periodic crystalline "sheet". By crystalline I simply mean that it is preserved by the action of integer ...
- 13.5k
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votes
2
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Exotic smooth structures on Lie groups?
If a topological group $G$ is also a topological manifold, it is well-known (Hilbert's 5th Probelm) that there is a unique analytic structure making it a Lie group.
However, for a compact Lie group $...
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A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$
Edit: According to the comment by @LSpice we realise the existing link to the main motivation of the question is not available. Then we search for the paper we found the following version:
https://www....
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1
answer
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Universal Chevalley group associated to $D_l$
Consider the simple Lie algebra $D_l$. Consider the universal Chevalley group $G$ over a field $K$ associated to it. Then $G$ is a subgroup of the orthogonal group $O_{2l}(K, f)$ where $f$ is the ...
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votes
1
answer
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Hua Luogeng's definition of automorphism group for Hermitian symmetric space
I'm trying to make sense of a definition appearing in Hua Luogeng's book "Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains".
Consider the Hermitian symmetric space ...
2
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answers
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A question about the associative classes of parabolic subgroups
Let $\mathbb{A}$ denote the adele group of $\mathbb{Q}$. Suppose that $P_1$ and $P_2$ are parabolic subgroups of a reductive algebraic group $G$, and consider their Langlands decomposition
$$
P(\...
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Pedagogical question on Lie groups vs. matrix Lie groups
There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
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votes
1
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Maximal torus of Chevalley group $Sp(4)$
Consider a chevalley group a field $K$, with the right chevalley basis. Let $\alpha$ be a root. Let $x_{\alpha}(t)$ be the corresponding root space. Define $w_{\alpha}(t)=x_{\alpha}(t)x_{-\alpha}(-t^...
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answers
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What is known about Lie groups with (strictly) positive curvature?
If we consider $G$ a compact Lie group, there is a left invariant Riemannian metric whose the sectional curvature is nonnegative (see Milnors' paper). When can we find a left invariant metric that has ...
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votes
1
answer
645
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Grothendieck's paper on principal bundles, reduction to a torus step
In Grothendieck's paper "Sur la Classification des Fibres Holomorphes sur la Sphere de Riemann", there is a step I don't understand in section 4, where he proves reduction to a torus. He states (lemma ...
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votes
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answers
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classification of homogenous complex manifolds
Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?
- 1,091
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Milnor's model of $EG$ and Kac-Moody groups
I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 ...
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votes
1
answer
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Homology of solvable Lie groups made discrete
In what follows "homology" will mean group homology, i.e. $H_*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology.
It is well-known how to compute the homology of abelian groups, ...
- 10.3k
6
votes
1
answer
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Are the integer matrices in SO(3,2) "boundedly generated"?
Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$.
(The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric ...
- 11.2k
6
votes
2
answers
695
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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}(\...
- 5,933
5
votes
1
answer
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Triviality of a fiber bundle
Is the principal fiber bundle $GL^+(6,\mathbb R)$ over $GL^+(6,\mathbb R)/SL(3,\mathbb C)$ trivial ?
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