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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A class of Lie groups with $f^{abc} \neq -f^{acb}$ (not fully anti-symmetrized) or $f^{abc} \neq f^{bca}$ (not-cyclic)
With the motivation to understand the Lie group structure constraint on a non-Abelian Chern-Simons theory, could some experts give a class of Lie groups with structure constants cannot fully anti-...
- 81
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1
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Can we prove that there are countably many isomorphism classes of compact Lie groups without the classification of simple Lie algebras?
This is an old math.SE question of mine that was never answered:
It is a nontrivial fact that there are only countably many isomorphism classes of compact Lie groups. One can prove this by a series ...
- 118k
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1
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Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a (pseudo)Kähler-Hodge manifold? Open problem
I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer.
Is the manifold
$$M=\frac{E_{7(7)}}{SU(7)}\times \mathbb{...
- 2,818
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votes
2
answers
973
views
maximal compact subgroup as fixed points of some involution on p-adic group?
As is well known, maximal compact subgroup of real Lie group is just the fixed points of Cartan involution.
Now the question is what's the possible p-adic analog?
- 2,709
3
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1
answer
355
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Indefinite orthogonal groups over p-adics
Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...
- 1,692
3
votes
1
answer
98
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Locally nilpotent algebraic section of tangent bundle is complete?
Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...
user75660
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1
answer
529
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Cotangent bundle of symmetric space is symmetric space?
Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...
user21574
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0
answers
95
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what kind of Gaussian matrix models are these?
In a physics paper I found a very complicated Gaussian matrix model:
$$ Z = \int \frac{d\mu}{(2\pi)^n} \frac{d\nu}{(2\pi)^n}
\frac{
\prod_{i < j}\left[2 \sinh \frac{\mu_i - \mu_j}{2} \right]^2 \...
- 22.8k
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301
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Geometric proof of Borel-Weil theorem
I am curious if there is any geometric proof of Borel-Weil theorem.
Borel-Weil is a geometric realization of irreducible unitary representation. The proofs I found, however, all use Weyl unitarian ...
- 587
2
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0
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143
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Diagonal invariants of $SO(n)$
Consider a Lie algebra $\mathfrak g$ (I am mostly interested in the case $\mathfrak g=so(n)$), its universal enveloping algebra $U$ and its center $C$. There is an adjoint action of $\mathfrak g$ on $...
- 1,488
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1
answer
97
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Zariski open set in orthogonal grassmanian [closed]
I am confused about the following question.
Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&...
- 1,121
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votes
1
answer
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Haar measure on infinite dimensional Lie groups?
Hi. Is there a Haar measure or equivalent on infinite dimensional Lie groups? I've been playing around with $Diff(S^1)$, and at least a direct approach seems quite hopeless. It goes something like ...
- 548
3
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0
answers
267
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adding a boundary to the finite upper half-plane
Let $\Bbb{F}_q$ be a finite field, let $\delta \in \Bbb{F}_q$ be a non-square, let $\Bbb{F}_{q^2} = \Bbb{F}_q\big( \sqrt{\delta} \big)$ be the corresponding quadratic extension,
and let ${\frak{H}}_q:=...
- 2,137
14
votes
1
answer
503
views
Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent?
I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group.
Is it true? Where can I find a proof?
A counterexample for open subsets of $...
- 2,479
6
votes
1
answer
343
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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, ...
- 11.2k
3
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0
answers
126
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Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra
Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k \mathfrak{h}...
- 159
1
vote
1
answer
1k
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Representation of quotient group
Hello,
My question is about the relation between representations of a Lie group and its quotient.
Let $G$ be a compact lie group and $H$ a central subgroup of $G$. What is the relation between ...
- 129
1
vote
0
answers
159
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Notation clash between a representation and spectral radius
I am currently writing a paper where I need talk both about a representation of a semisimple Lie group (usually denoted by $\rho$), and about spectral radii of linear maps (also usually denoted by $\...
- 1,574
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votes
1
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142
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What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite
When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
- 1,040
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votes
2
answers
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Christoffel symbols on a Lie group in Riemann normal coordinates
Consider a coordinate patch around the identity element in a Lie group given by the exponential mapping (Riemann normal coordinates). We have a Levi-Civita connection corresponding to the bi-...
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0
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955
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Injectivity of Lie group exponential function
If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and $\...
- 1,040
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0
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194
views
What are the E7(7) invariants in the adjoint representation?
Take a real vector space $R$ transforming in the adjoint representation of
the ${\rm E}_7(7)$ Lie group as $R \rightarrow G R G^{-1}$. One can define
invariants using traces of products of $R$ as ${\...
4
votes
1
answer
256
views
Weyl group action on complexified Iwasawa decomposition
Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...
- 4,902
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0
answers
1k
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How to find the unitary matrices in this exponential matrix representation
In the following post
Representing a product of matrix exponentials as the exponential of a sum
there is a statement regarding the result of the multiplication of two matrix exponentials:
if $A$ and $...
- 149
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votes
0
answers
161
views
Intermediate quotient for a Hermitian Symmetric Spaces of $Sp(n)$
We know that $U(N)$ can be embedded into $SU(n+1)$ and that the quotient is isomorphic to complex projective space:
$$
SU(n+1)/U(n) \simeq {\mathbb CP}^{n}.
$$
We can split this process into two ...
- 167
1
vote
0
answers
189
views
Poincaré inequality for connected Lie groups
Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:
symmetric,
adapted (in the sense that there is no proper subgroup $H$ such ...
- 527
0
votes
1
answer
208
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on lifting extensions
Let $G$ be a connected reductive group with $G_{der}$ simply connected and $T$ a maximal torus over an algebraically field $k$.
We consider a extension $\tilde{T}$ of the maximal torus $T$ by a torus ...
- 3,472
5
votes
1
answer
653
views
Reference request for the list of maximal subgroups of SU(3,1)
Is there a reference with the list of maximal subgroups of SU(p,q) for "small" values of p and q? (such as SU(3,1) as suggested in the title of the question)
- 1,675
4
votes
1
answer
715
views
Differential equations and Lie groups
I am a physicist and I am pondering over a particular generalization of Stokes' theorem and Maxwell's equations. They apply to vector fields like the electric or magnetic one. However if the vectors ...
1
vote
0
answers
100
views
Representation equivalent lattices
Suppose $G$ is a absolutely almost simple algebraic groups over a number field $K$. Let $\Gamma_1$ and $\Gamma_2$ be two lattices in $G(K)$. Then $\Gamma_1$ and $\Gamma_2$ are said to be ...
- 61
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0
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135
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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
7
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1
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424
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Open cell decomposition after applying a Weyl group element
Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step.
For Zariski-...
- 4,902
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vote
1
answer
215
views
Relation between volume of reduced space and phase space
Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is $$...
user21574
2
votes
1
answer
364
views
Variant of the Riemann Mapping Theorem for $Conf(\mathbb H^2)$?
According to the Riemann mapping theorem it is possible to map a simply connected open subset $B \subset \mathbb C$ into any other $B' \subset \mathbb C$ by a (bi-)holomorphic mapping. Moreover, such ...
- 548
1
vote
1
answer
250
views
family of metrics with same geodsics
For every bi-invariant metric on a lie group we know geodesics are flow of left invariant vector fields, so this question naturally arise:
are there family of metrics on manifolds that have same ...
- 327
2
votes
0
answers
112
views
Eigenvalues of the imaginary part of the Symplectic action on Siegel upper half plane
Let $A,B\in M_n(\mathbb{R})$ and $U=A+iB$ unitary. $R=diag(r_1,r_2,…,r_n)$ is a diagonal matrix with $r_i>0, \forall i $. I need to calculate $\det(Ae^{-R}A^T+Be^{R}B^T)$. This matrix $Ae^{-R}A^T+...
- 31
7
votes
1
answer
426
views
Lie algebra "generated" by matrix-valued curve?
Let $A(t)$ be a $n\times n$-matrix-valued continuous (plus possibly other niceness conditions; see below) curve, with the matrix entries being complex in general. If I am not mistaken, $A(t)$ ...
- 784
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428
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Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
- 181
3
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0
answers
196
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Uniform sub-linearity of sub-additive functions on groups
Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...
- 1,101
3
votes
2
answers
409
views
Connectedness of Springer Fibers
Let $G$ be a connected, simply-connected, complex semisimple Lie group with Lie algebra $\frak{g}$. Let $\mu:T^*\mathcal{B}\rightarrow\mathcal{N}$ be the Springer resolution of $\mathcal{N}$. If $G=\...
- 4,920
4
votes
1
answer
146
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connections on principal bundles over $S^1$
Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My ...
- 43
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0
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400
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A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?
The following is my first question here on mathoverflow.
Let $M$ be a closed connected Riemannian manifold with an isometric effective action of a compact connected Lie group $G$. Consider the ...
- 1,942
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votes
2
answers
513
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Sample from a delta-ball in the orthogonal group O(n)
An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ...
- 201
0
votes
1
answer
587
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fiber bundle on an orbit of $\mathfrak{g}\oplus\mathfrak{g^*}$
Let $G$, be a Lie Group and $\mathfrak{g}$ be its Lie algebra ,i.e, $Lie(G)=\mathfrak{g}$. Let $\zeta=(\ X,F)\ \in \mathfrak{g}\oplus\mathfrak{g^*}$. Here $X\in \mathfrak{g} $ and $F\in \mathfrak{g^*}$...
user21574
6
votes
1
answer
988
views
Lie algebras and complements
I have some elementary questions about Lie algebras and vector space complements.
Let $(\mathfrak{g},[.,.])$ be a finite-dimensional Lie algebra and $\mathfrak{g}_1$ a Lie ideal in $\mathfrak{g}$.
...
- 1,222
2
votes
0
answers
115
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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 ...
- 2,099
2
votes
1
answer
143
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Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$
Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which:
$K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...
- 2,099
1
vote
1
answer
609
views
Para-Complexification of Lie Groups
Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...
user21574
2
votes
1
answer
286
views
The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$
Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let $...
user21574
6
votes
1
answer
298
views
Invariants of a $GL(3,\mathbb{R})$ action
I'm trying to understand the standard $GL(3,\mathbb{R})$ action on the 15-dimensional space of possible values for the derivative of the Riemann curvature tensor of a 3-dimensional manifold $M$ at a ...
- 818