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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Infinite-dimensional admissible representations of SL(2,C)
I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...
0
votes
2
answers
446
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Classifying compact homogeneous Kähler manifolds
In this comprehensive answer to an old question, it is stated that
Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.
...
- 163
2
votes
1
answer
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Generalization of the Lie group exponential map and its derivative
Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$, and $exp:\mathfrak{g}\to G$ be its exponential map. The group $G$ could be finite or infinite dimensional. Let $G$ have the property that
$\...
- 1,467
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votes
1
answer
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Classification of finite-dimensional continuous irreps of affine group up to isomophism?
Let $k$ be a field. For each $a \in k^\times$ and each $b \in k$, let $g_{a, b}: k \to k$ be an affine-linear map given by $g_{a, b}(x) = a \cdot x + b$. The transformations $\{g_{a, b}, \text{ }a \in ...
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compact almost complex submanifolds of complex Lie groups
Does there exist any complex Lie group $G$ such that there are some positive-dimensional compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$?
I want to get some examples.
...
- 2,223
4
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How to tell when two abstract root data are isomorphic
This is a related question to one I just asked ($\textrm{GSp}_{4}^{\wedge} \cong \textrm{GSp}_4$).
Let $\Psi=(X,R,X^{\wedge},R^{\wedge}), \Psi_1 = (X_1,R_1,X_1^{\wedge},R_1^{\wedge})$ be two root ...
- 6,180
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answer
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Kähler form on complex Lie group
Hallo,
Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G_{\mathbb{C}}$. Does there exist a Kähler structure on $G_{\mathbb{C}}$ which is $G$-invariant (maybe in a ...
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Simple discrete subgroups of Lie groups
Upon Ian Agol's suggestion, I separated this question from the one on non-residual finiteness in
Non-residually finite matrix groups
Question. Are there infinitely generated simple discrete ...
- 31.2k
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votes
2
answers
418
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About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$
Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) $...
- 9,019
13
votes
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Homogeneous spaces that are homotopy tori
Let $G$ be a compact Lie group, and let $H$ be a closed subgroup such that $G/H$ is homotopy equivalent to a torus. Is it true that $H$ is normal and $G/H$ is isomorphic to a torus as a Lie group?
...
- 56.9k
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How large is the intersection of the root system of a subalgebra of a compact Lie algebra with the original root system?
Let $\mathfrak{g}$ be a finite-dimensional real compact Lie algebra and $\mathfrak{t}\subset \mathfrak{g}$ a maximal abelian subalgebra. Let $\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})\...
- 1,942
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Submanifolds of nilmanifolds coming from Lie subgroups
Let $G$ be a connected simply connected nilpotent real Lie group and $\Gamma$ a lattice in $G$, such that $M=\Gamma \backslash G$ is a compact nilmanifold. Let $p:G \to M$ be the projection. If $S$ is ...
- 209
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Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and $S^-...
- 1,719
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answer
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Does a Trivial Tangent Bundle Induce a Multiplication?
Let $M$ be a connected smooth manifold, and assume that it is parallelisable; that is, its tangent bundle is trivial. Does $M$ admit an H space structure? That is, does there exist a smooth map $\mu:...
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Frame-bundle reduction from spinor-bundle reduction
Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times_{\rho}\Delta$ its spinor bundle, ...
- 2,818
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0
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Identification of spectral and differential data for integrable difference equations?
Let $X$ be a projective curve and $G$ be a semisimple Lie group. There is a theorem roughly stating that there exists an isomorphism between the moduli space of principal $G$-bundles on $X$ and the ...
user74900
3
votes
2
answers
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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: \...
- 2,137
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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 ...
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Borel's Paris Lectures
I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or ...
- 105
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votes
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462
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R-linear representations of sl(2,C)
Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$?
Equivalently, what ...
- 10.4k
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Hamiltonian Group action with infinitely many stabiliser types
What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types?
Infinitely many stabiliser types means that ...
- 6,995
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Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 2,097
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3
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752
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How can I tell whether a manifold is homogeneous?
I have been influenced by this question with many beautiful answers.
Are there any useful practical criteria to say positively that a real connected paracompact smooth manifold $X$ is homogeneous?
I ...
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3
votes
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Does convergence in orbit imply convergence in group for finite stabilizer?
Let $G=\operatorname{GL}_n(\Bbb C)$ act polynomially on some finite-dimensional complex vector space $V$. This means that the action is given by a morphism $\rho\colon G\to\operatorname{GL}(V)$ of ...
- 4,902
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0
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Which operators constructed from 10d gamma matrices commute with $SO(1,2)\times SO(3)\times SO(3)$?
In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in ...
- 1,155
13
votes
3
answers
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Nearby homomorphisms from compact Lie groups are conjugate
I'm looking for a proof (that I can understand) of the following fact: If $K$ and $G$ are Lie groups, and $K$ is compact, then nearby homomorphisms $K\to G$ are conjugate.
That is, if $\mathrm{Hom}(...
- 27.2k
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Connections on a Lie Group
A Lie group $G$ can be considered as a reductive homogeneous space in at least two different ways; $G/\{e\}$ and $G\times G/G^*$. In the first case, the canonical connection associated with the ...
- 1,378
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2
answers
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Borel–Weil theorem - reference request
I am asking about good references (both books and papers) for the well-known Borel–Weil theorem. Thank you very much!
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Fibre bundles and flat connections [closed]
If a fibre bundle can be equipped with a flat connection then it must be necessarily trivial? Let us take for example a real line bundle $L\to M$ with base $M$. If $L$ can be equipped with a flat ...
- 2,818
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1
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Why a nilpotent Lie group must be a matrix group?
The question may be a little naive (or even appear as a duplicate) as I guess the result is well known. I saw on the other thread that
"
c) A solvable Lie group G is linear iff its commutator ...
- 6,259
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votes
1
answer
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unitary irreps of O(p,q)
I am interested in the irreducible unitary representations of the orthogonal groups $O(p,q)$. By $O(p,q)$ I mean the real Lie groups which preserve the quadratic form of signature $(p,q)$ in $\mathbb{...
- 295
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votes
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Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold
Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing volume-...
- 277
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2
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Closure relations between Bruhat cells on the flag variety
Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...
- 1,719
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1
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Getting the story of Dynkin and Satake diagrams straight
I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...
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1
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Faithful representation of the projective unitary group with the lowest dimension?
What is the lowest dimension of a faithful ordinary representation (as compared with projective representation) of the projective unitary group $\rm{PU}(d)$? Is it $d^2-1$?
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Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?
Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows:
$$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
- 499
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votes
1
answer
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Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $
My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...
user21574
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0
answers
399
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Is the Lie derivative of a harmonic form also a harmonic form?
On Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces" it is said that the Lie derivative along a left-invariant vector field of an harmonic form is again a harmonic form. This ...
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Fubini--Study Orthogonality for Schubert Calculus
Consider the following points:
$\bullet$ Let ${\cal Harm}(n,d)$ denote the harmonic forms of the de Rham complex of the Grassmannian $Gr_{\mathbb{C}}(n,d)$ with respect to the Riemannian metric ...
- 443
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1
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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 ...
- 6,025
3
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1
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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, ...
- 10.4k
13
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3
answers
3k
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Representations of \pi_1, G-bundles, Classifying Spaces
This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first ...
- 2,684
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0
answers
193
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Non-existence of nontrivial finite group extension of any simply-connected Lie group
Let $Q$ be a simply-connected compact Lie group. Can one outline the proof (or provide the counter examples if my statement is false) that
there does not exist any group $G$ (with no topology) ...
- 10.5k
3
votes
1
answer
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$SO(N^2-1)$ and the adjoint representation of $SU(N)$
It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$.
I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...
- 315
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1
answer
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When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?
This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected.
It seems that the ...
- 5,565
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votes
1
answer
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Learning representation theory of real reductive lie groups
I am interested in any sources that can be helpful for learning the representation theory of real reductive groups.
I am currently reading Wallach book, but I feel that I don't understand the subject ...
6
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1
answer
255
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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, \...
- 6,211
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1
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332
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Orientability of orbit type strata of Lie group actions
Let $G$ be a compact Lie group that acts on a smooth, finite dimensional, oriented manifold $M$, and suppose that such action preserves orientation, i.e., for each $g\in G$, the diffeomorphism $\mu_g$ ...
- 51
1
vote
2
answers
2k
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The normalizer of a reductive subgroup
Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the ...
- 4,280
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1
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809
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How does one calculate homotopy classes for group coset spaces?
Inspired by Witten's Wess-Zumino term arguments, I'm curious to know how one calculates homotopy classes more generally for coset spaces. In the above example the coset is $G/H=(SU(3)_L\times SU(3)_R)...
