Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
1,849 questions
3
votes
0answers
82 views
Examples of incomplete Lorentz 3-manifolds
Reading this paper where closed 3-dim. Lorentz manifolds with noncompact isometry groups are studied, I wonder if all of them are geodesically complete.
One class of 3-dim. closed Lorentz manifolds ...
8
votes
1answer
133 views
Flag manifolds as incidence correspondences
Let $G$ be a reductive group, $B$ a Borel and $P_j$ the maximal parabolics, indexed by the vertices $j$ of the Dynkin diagram. Then $B = \bigcap_j P_j$, so the flag manifold $G/B$ injects into $\...
3
votes
0answers
33 views
Commutator length in connected solvable Lie groups
Let $G$ be a connected solvable Lie group and let $H$ denote ist commutator subgroup. By definition, every element $g \in H$ can be written as a product of commutators and the minimal number of ...
0
votes
0answers
50 views
Is conjugation by gauge transformation of $G$-bundle contained in $\mathfrak{g}$?
The following question arises from Part II, Exercise 86 of Gauge fields, knots, and gravity by Baez and Muniain.
Let $M$ be a smooth manifold, let $G$ be a Lie group, and let $\pi\colon E\to M$ be a $...
13
votes
1answer
211 views
What is the value of $[S^3/G] \in \pi_3(Sphere)$ for a finite subgroup $G \subset SU(2)$?
Let $G\subset \mathrm{SU}(2)$ be a finite group. (These are famously classified through the McKay correspondence.) The Lie group framing of $\mathrm{SU}(2) = S^3$ descends to the quotient manifold $S^...
7
votes
0answers
100 views
Can semisimple orbits be written $\exp(\mathfrak{g})\cdot x$?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit
$$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$
...
3
votes
1answer
139 views
Is $({F^{\times})^{ diag}}\backslash(GL_2 \times E^{\times})_{det=\mathbb{N}}$ a unitary group?
Let $F$ be an p-adic field, and $E$ be a quadratic extension of $F$, then is $({F^{\times})^{ diag}}\backslash(GL_2 \times E^{\times})_{det=\mathbb{N}}$ isomorphic to some unitary group $U_{E/F}(2)$? ...
0
votes
1answer
84 views
Fibered product of stacks comes from a Lie groupoid
Suppose $\mathcal{G},\mathcal{H}$ are Lie groupoids and $F:B\mathcal{G}\rightarrow B\mathcal{H}$ be a morphism of stacks.
We can talk about the fibered product $B\mathcal{G}\times_{B\mathcal{H}}B\...
7
votes
2answers
232 views
Infinite Krull-Schmidt categories?
In a Krull--Schmidt category, if
$$
X_{1}\oplus X_{2}\oplus \cdots \oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{s},
$$
where the $X_{i}$ and $Y_j$ are all indecomposable, then $r = s$, ...
2
votes
0answers
142 views
Reference for a proof of a Theorem by Joseph Wolf
We know that Lie Groups are parallelizable, I was looking for a version of the converse and came across this:
https://books.google.com/books?id=w4bhBwAAQBAJ&pg=PA115 in Introduction to Smooth ...
1
vote
0answers
67 views
Embedding of discrete series
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals that of its maximal compact subgroup. Let $G'$ be a reductive subgroup of $G$ with equal rank. If $\tau$ is a discrete ...
1
vote
0answers
66 views
Coordinate ring of an equivariant embedding of a homogeneous projective variety
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...
7
votes
1answer
116 views
Lie algebra preserving ideal of functions
Let $G$ be an algebraic group acting on an affine variety $X=\operatorname{Spec}A$ (all over $\mathbb{C}$). This gives an action of $G$ on the $\mathbb{C}$-algebra $A$, and an action of the Lie ...
5
votes
0answers
74 views
Group cohomology of “twisted” projective SU(N) with various coefficients
Given a group
$$
G= PSU(N) \rtimes \mathbb{Z}_2,
$$
where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then
$$
c a c= a^*,
$$
which $c$ flips $a$ to its ...
3
votes
0answers
94 views
+50
Structure of a group acting on a Hilbert space
Assume a group $G$ acts faithfully by isometries on a separable infinite dimensional Hilbert space $H$ in such a way that the orbits are closed and the quotient $H/G$ is isometric to a finite ...
8
votes
0answers
216 views
Significance of half sum of non-simple positive roots
In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
7
votes
1answer
172 views
Polynomial invariants for simple algebraic groups
Let $G$ be a simple complex algebraic group. Let $V$ be a finite-dimensional algebraic representation of $G$. Thus, we can write $V=V_1\oplus \cdots \oplus V_n$ where $V_i$'s are irreducible ...
0
votes
0answers
140 views
The order of the map $\Sigma^3 SU(4) \to SU(4)$
What is the order of map $\alpha:\Sigma^3 SU(4) \to SU(4)$ where $\Sigma^3$ is three-suspension. Let $A$ be the order of this map, I must show that $A = 60$. We know that $A$ is equal to $120$ or $60$,...
3
votes
1answer
116 views
Invariant integration on principal bundles
Let $G$ be a sufficiently nice topological or Lie group (e.g. compact), and let $H$ be a closed subgroup. This data determines a principal $H$ bundle $G \rightarrow G/H$ defined by the projection $g \...
7
votes
0answers
69 views
Normalizers of subsystem subgroups of Lie groups
Let $G$ be a semisimple complex Lie group, and let $H$ be a subgroup corresponding to a subset of the extended Dynkin diagram of $G$ (à la Borel - de Siebenthal). I would like to know if there is a ...
4
votes
1answer
197 views
Flat solvmanifolds?
I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds ...
1
vote
2answers
210 views
Examples of group $G=N \rtimes H$ where $N$ and $H$ are as below
I am searching for examples of connected locally compact group $G = N \rtimes H$, where $N$ is a simply connected nilpotent non-abelian Lie group, $H$ is linear reductive and $H$ operates on $N$ ...
2
votes
1answer
130 views
Relative weight lattice
Let $G$ be a reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus, $B$ be a Borel subgroup and $I_G$ is the set of simple roots. Let $P$ be a standard parabolic subgroup, ...
2
votes
0answers
88 views
Invariants of the group $SO(2)$
Let $V_d$ be the complex vector space of binary forms of degree $d$ endowed with the natural
action of the special orthogonal group $SO(2).$ Consider the corresponding action of the
group $SO(2)$ on ...
3
votes
1answer
204 views
Homotopy type of $SO(4)/SO(2)$
A classical result states that the quotient $SO(4)/SO(3)$ is homotopy equivalent to $S^3$. In fact, this can be stated in more general terms since $SO(n+1)/SO(n)$ has the homotopy type of $S^n$. What ...
7
votes
1answer
148 views
Finite subgroups of $PSU(3)$
I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
2
votes
0answers
55 views
Finite-dimensional graded Lie algebras with $2$ generators
Does anyone know of a classification of those (complex) Lie algebras which are:
generated by two elements
$\mathbb{Z}$-graded Lie algebras
finite dimensional
6
votes
1answer
141 views
Irreducibility of Gelfand-Serganova strata
To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $...
11
votes
0answers
106 views
Intrepreting Spin(3) as a certain configuration space
Let $\mathcal{C}$ denote the space of great circles in $\mathbb{S}^2\subset \mathbb{R}^3$. It's pretty easy to see that any element $\mathcal{C}$ can be identified uniquely with the axial line (in $\...
7
votes
1answer
399 views
Stabilizer of Sp(n) and U(n) in GL(n)
I would be very grateful for a reference
to the following results (which are, I think, true,
though I never saw it in the literature).
Let $G\subset GL(n,{\Bbb C})$ be $U(n)$,
abd $A\in GL(2n,{\Bbb ...
19
votes
0answers
234 views
The de Rham complex of the octonionic projective spaces
The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...
0
votes
0answers
80 views
Vector-valued forms in Riemannian geometry
Suppose $(M,g)$ is a Riemannian manifold. I want to find a vector-valued $2-$form
$T$ such that, for any vector fields $X,Y,Z$ on $M$,
$$
g(T(X,Y),Z)=g(T(Z,X),Y)\,.
$$
As a motivation, consider the ...
7
votes
0answers
135 views
Which representations of the Lie algebra of a Lie group come from representations of the group itself?
I think this is very classic mathematics, but I can't find a complete answer in the literature.
Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
3
votes
0answers
56 views
Maximally symmetric affine manifold
As a physicist who knows (something) about General Relativity, I'm accustomed to the term "maximally symmetric space" being an $n$-dimensional manifold with $\frac{n(n+1)}{2}$ Killing vectors. A ...
4
votes
1answer
114 views
Integration of Maurer-Cartan form
Let $G$ be a Lie group with Lie algebra $g$. As it is well known the Maurer-Cartan form $ω:TG\rightarrow g$ transports any vector $X\in T_{x}G$ to the start $l_{x^{-1}*}(X)\in g$, $l_{x^{-1}}$ ...
8
votes
1answer
205 views
Integral of product of Schur functions
Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae
$$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^...
8
votes
2answers
309 views
Hyperbolic $3$-manifold groups that embed in compact Lie groups
Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...
4
votes
1answer
325 views
Classification of compact connected abelian groups
It is known that torsion-free compact abelian groups are exactly the product of the maximal solenoid group $\Sigma_{(2,3,\cdots)}$ (which is the Pontryagin dual of the additive group $\mathbb{Q}$ of ...
4
votes
1answer
86 views
Spin groups in terms of matrices and/or linear operators
Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...
8
votes
2answers
333 views
$G(k)/H(k)$ as a submanifold of $G/H(k)$
Let $k$ be a local field (if necessary, assume characteristic zero). In general, if $X$ is a smooth variety of finite type over $k$ of dimension $n$, then the set of $k$-rational points $X(k)$ is an ...
5
votes
1answer
112 views
The Hausdorff dimension of the union of singular orbits and exceptional orbits
Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...
8
votes
1answer
234 views
Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones
Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...
5
votes
0answers
113 views
Interesting Subgroups of $SU(2^n)$
I'm a physicist, so please forgive me if this question is wrong or simplistic, but are there any subgroups of $SU(2^n)$ which are isomorphic to $SU(k)$ for some $k \geq 2$, which do not have a non-...
6
votes
2answers
316 views
Fundamental representations and weight space dimension
For the Lie algebra $\frak{sl}_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all ...
4
votes
0answers
163 views
Chern-Simons theory with non-compact gauge groups G
This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...
3
votes
1answer
141 views
Littlewood Richardson Rule for general linear group over finite field
I just finished reading Green's 1955 paper on characters of general linear groups and have also been reading Macdonald's Symmetric Functions and Hall Polynomials. I see that there is a recursive ...
1
vote
0answers
30 views
Defining a notion of “volume of its lattice” for non-rational subspaces
Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”:
$$\...
4
votes
1answer
125 views
Nonlinear sigma models with non-compact groups / target spaces
A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T.
The target manifold T is equipped with a Riemannian metric g. Σ is ...
9
votes
0answers
151 views
When is G->G/H a trivial bundle
Suppose that $H\subseteq G$ are connected lie groups. Then $G\mapsto G/H$ is a principal $H$-bundle. I would like to know if there is some non-tautological characterization of when this is a trivial ...
31
votes
2answers
558 views
Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
