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
2
votes
1
answer
462
views
Split real form of $SL(2,\mathbb{C})$ description of the two sphere?
If we denote the parabolic subgroup of $SL(2,\mathbb{C})$ by $P$, then we have the well known isomorphism $SL(2,\mathbb{C})/P \simeq S^2$, where $S^2$ is the two sphere. Now the compact real form of $...
- 1,453
4
votes
0
answers
468
views
Complex symplectic reduction
Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...
- 1,347
3
votes
3
answers
461
views
Multiplicity of eigenvalues in 2-dim families of symmetric matrices
Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
- 1,452
1
vote
1
answer
274
views
The compact Lie group contains a finite subgroup $\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$
Given a finite Abelian group: $G=\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$, where ${n_1},{n_2},{n_3}$ are arbitrary positive integers. ${n_1},{n_2},{n_3}$ may have or may not ...
- 10.5k
1
vote
2
answers
465
views
subgroups with the same number of roots that the group.
When thinking in terms of Dynkin diagrams, I am naively used to see that the diagram for a subgroup can be extracted from the diagram of the group by removing some roots. Now, I noticed that for SO(10)...
- 437
3
votes
0
answers
362
views
Unitary representation of finite-dimensional unitary group
the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a ...
4
votes
2
answers
363
views
Complexification or 'real'ization of Mapping Class group.
So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
- 327
0
votes
0
answers
1k
views
Bi invariant Riemannian metric on a Lie Group
I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant ...
0
votes
0
answers
612
views
Left invariant connection on Lie groups
Let $M^n$ be a Riemannian manifold, $G^{n + k}, \: k\geq 1 $ be a connected Lie group equipped with a left invariant metric and $f:M^n \to G^{n + k}$ one isometric immersion. Let $X, Y$ left invariant ...
- 1
3
votes
2
answers
338
views
compute the automorphism of Iwasawa manifold
An Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup.
We can also refer to Griffiths and Harris's Principles of Algebraic Geometry p....
- 197
0
votes
1
answer
138
views
Gradient on $SU(n)$
I'm trying to calculate the gradient (wrt to the bi-invariant metric) of the following functions $F_1, F_2 : SU(n) \rightarrow \mathbb{R}$ defined by $F_1(U) = | Tr (G^{\dagger} U) |^2$, $F_2(U) = \Re ...
- 2,099
1
vote
1
answer
172
views
Existence of a Lie algebra element orthogonal to the adjoint orbit of another element
Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$.
Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) ...
- 2,099
3
votes
1
answer
614
views
Coadjoint orbits and homogeneous symplectic $G$-manifolds
We know this important fact from A.A.Kirillov that :
Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central ...
user21574
4
votes
1
answer
212
views
Subgroups of $Sp_{2g}$ giving rise to Shimura data
Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...
user70064
3
votes
0
answers
386
views
What are the general zonal spherical functions for ${\rm SO}(n)/{\rm SO}(n-1)$?
The zonal spherical functions [1] on the sphere $(G={\rm SO}(n)$, $K={\rm SO}(n-1))$ are the Gegenbauer or ultraspherical polynomials if one considers the irreducible representations of ${\rm SO}(n)$ ...
6
votes
2
answers
5k
views
Representations of Lorentz group
Questions:
What is the connection between representation theory of complex semisimple Lie groups and representations of (maybe "proper") Lorentz groups?
Why should one read Bargmann's paper on irred. ...
- 361
5
votes
0
answers
214
views
What do we know about isospectral Cayley graphs?
Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...
- 1,893
4
votes
2
answers
511
views
Equivariance of vector bundles over G/B
Let $G$ be a complex semisimple group, $B$ a Borel subgroup of $G$ and $X=G/B$ the flag variety of $G$. If $G$ is simply connected, then every line bundle $L$ on $X$ can be made $G$-equivariant (see ...
- 10.3k
2
votes
0
answers
232
views
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 ...
- 2,508
1
vote
1
answer
461
views
Subgroups with trivial Centralizers
Let $G$ be a compact, simply-connected, and simple Lie group. Let $H$ be a closed subgroup of $G$ that has the same centralizer as the center of $G$. Is there a nice classification of such subgroups?
...
- 19
2
votes
1
answer
259
views
$\text{mod} \, p^2$ trace identity
Let $p$ be a prime, and let $\text{GL}_n \big( \Bbb{Z} / p^2 \Bbb{Z} \big)$ be the group of $n \times n$ invertible matrices over the ring $\Bbb{Z} / p^2 \Bbb{Z}$. Does there exist a positive integer $...
- 2,137
1
vote
1
answer
233
views
Non Hamiltonian vector field
Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group.
Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...
- 181
7
votes
1
answer
561
views
How does the right regular of GL(n, R) and GL(n,Qp) decompose?
The question is contained in the title. I would guess that this question is already answered in the literature.
Given the reductive group $GL(n)$ over a complete local field, how does the right ...
- 11.2k
1
vote
1
answer
301
views
Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure
I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...
user21574
9
votes
1
answer
804
views
Local trivializations of the non-trivial $SU(2)$-bundle over $S^5$
It is well known that $SU(3)$ is the unique, non-trivial, principal $SU(2)$- bundle over $S^5$. To my knowledge the way this is proven is by using the following fact:
If $G$ is a Lie group and ...
- 627
2
votes
1
answer
370
views
Relationship between Verma modules and delta functions
Reposting my question from math.stackexchange:
What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory (http://www.math.columbia.edu/~woit/...
- 441
2
votes
1
answer
367
views
Lie group action with no slice
Let $x\in M$, $M$ - finite dimensional smooth manifold. Is there an example of a finite dimensional Lie group action on $M$ with no slice at $x$?
- 58
3
votes
0
answers
404
views
rational representation of semisimple algebraic group
Let $G$ be a connected semisimple algebraic group defined over $\mathbb Q$. Could some expert give me a complete classification of finite dimensional $\mathbb Q$-irreducible representations of $G$?
...
- 853
1
vote
1
answer
303
views
Bi-invariant one forms on compact Lie groups
I'm hoping to learn of a criterion for the existence of a bi-invariant one form on a Lie group $G$. I'm looking for a reference that there are no such one forms son $SU(n)$ (as long as this is in fact ...
- 2,099
4
votes
1
answer
381
views
The existence of a finite dimensional Lie algebra with a given symmetric invariant metric
The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
- 755
6
votes
2
answers
2k
views
Lie Semigroups?
Why is a Lie group wanted instead of a semigroup, what does the group structure give? References on this would be much appreciated.
I'm currently pondering manifolds and lie groups and their ...
- 1,801
3
votes
1
answer
77
views
Criterion for convergence of sums for non-continuous functions
The following question came up when thinking about equidistribution of Satake parameters of elliptic curves. Let $G$ be a compact Lie group with Haar measure $\mathrm{d} x$. Recall that a sequence $\{...
- 5,759
8
votes
2
answers
1k
views
number of irreducible representations over general fields
For a finite group, there are finitely many irreducible representations of complex numbers.
What if the field is changed to some other fields? Like real numbers, p-adic field, finite field?
In ...
- 1,503
8
votes
1
answer
1k
views
Symmetric tensor product of bosonic/fermionic Hilbert space
Consider two representation of the group $SU(n)$: $Sym^k(\mathbb{C}^n)$ and $\wedge^k\mathbb{C}^n$ ($k\leq n$) and take their symmetric tensor products: $Sym^2(Sym^k(\mathbb{C}^n))$, $Sym^2(\wedge^k\...
- 965
1
vote
1
answer
147
views
Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$
I have the following questions:
Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra $U(...
- 55
1
vote
1
answer
499
views
Existence of a fixed-point free map in a manifold [closed]
I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map.
I know ...
- 19
4
votes
0
answers
212
views
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 ?
- 1,121
2
votes
1
answer
403
views
surjective homomorphism with compact kernel (Milne's note on Shimura varieties)
I'm reading Milne's Introduction to Shimura varieties (http://www.jmilne.org/math/xnotes/svi.pdf) and there is something I don't get.
Let $G$ be a connected semisimple algebraic group $G$ over $\...
- 107
4
votes
1
answer
976
views
Character determines the representation?
Consider a semisimple Lie group or a $p$ adic reductive group $G$.
To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?
- 11.2k
1
vote
0
answers
106
views
Semiflows and continuous symmetries
Given a differential equation on a Banach space $\mathcal{X}$ of the form $\frac{d u}{d t} = F(u)$, it is often the case that $F$ is equivariant under translations, i.e. that $T_\alpha F(u) = F(T_\...
- 268
2
votes
1
answer
1k
views
Are certain simple Lie groups linear algebraic groups?
Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).)
Such a group should automatically ...
- 196
6
votes
1
answer
930
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 ...
- 687
3
votes
0
answers
51
views
Maximizing a function on $SU(4)$ similar to Von Neumann Trace Inequality
Given arbitrary $X,Y \in \mathfrak{su}(4)$, I want to maximize either of the following functions:
$\max_{U,V \in SU(2)} \Re(\text{Tr}(X^\dagger (U^{\dagger} \otimes V^{\dagger})Y (U \otimes V)))$
...
- 2,099
5
votes
0
answers
291
views
Hyperplane sections of principal homogeneous spaces
Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, ...
- 1,479
11
votes
1
answer
495
views
Is there a version of supersymmetry for homogeneous spaces?
The notion of "supersymmetry" that I am aware of proceeds as follows. One fixes a spacetime $\mathbb R^n$ and signature; I will write $\mathrm{SO}(n)$ for the corresponding group of orthogonal ...
- 54.6k
6
votes
1
answer
2k
views
A request for suggestions of advanced topics in representation theory
Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.
Let me begin by stating that I understand representation theory is a vast ...
5
votes
1
answer
453
views
A subgroup of the Weyl group
Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2.
Let $Q=Q(D)$ denote the root lattice of $D$.
Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is ...
- 14.2k
6
votes
1
answer
674
views
Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups
I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...
- 3,399
3
votes
1
answer
355
views
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
2
votes
1
answer
226
views
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