Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
-3
votes
0answers
15 views
Differential operators on $G/K$
Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal ...
1
vote
0answers
15 views
Differential operators on $G/K$
Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal ...
5
votes
0answers
102 views
What kind of locally symmetric space is a rational sphere
Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...
4
votes
1answer
136 views
Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)
It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...
0
votes
0answers
68 views
A lie group which is sat in its Lie algebra
Motivated by this question we ask the follwing question:
Assume that a Lie subgroup $G$ of $Gl(n,\mathbb{R})$ is contained in a subvector space $F$ of $M_n(\mathbb{R})$ such that $dim G=...
17
votes
1answer
370 views
Probability of satisfying a word in a compact group
This question is inspired by Probability of commutation in a compact group, which asked whether $P(xyx^{-1}y^{-1} = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. ...
2
votes
2answers
267 views
Maps from 2-Torus to SO(3)
Could someone please point me to a reference for topologically nontrivial maps from 2-Torus to SO(3), and how they are classified? [I'm a physicist, so a simple explanation would be useful]
5
votes
2answers
138 views
Generalizing Polar Decomposition of Matrices
I am trying to find a certain proof of polar decomposition of complex matrices which I think should exist more generally for a certain class of Lie groups. Recall that the polar decomposition of a ...
4
votes
0answers
240 views
Bézout and products in algebraic groups
Let $G$ be a linear algebraic group -- be it a Lie group or a group of Lie type. Let $V$, $W$ be subvarieties of $G$. Of course, $V\cap W$ is also a variety (not necessarily irreducible) and $V\cdot W^...
0
votes
0answers
38 views
Localizing a Clebsch-Gordan expansion around one representation
For the Lie group $\mathrm{SU}(2)$ the irreducible representations $\pi_m$ are labelled by non-negative integers $m$ and have dimension $(m+1)$. By the Peter-Weyl theorem, they form a basis for $L^2(\...
13
votes
3answers
709 views
Probability of commutation in a compact group
It is well known that if $G$ is a finite group, then the probability that two elements commutte is either $1$ (if $G$ is abelian) or less than or equal to $\frac58$.
If instead $K$ is a compact group,...
7
votes
0answers
142 views
Does Deligne's exceptional series lead to an “exceptional K-theory”?
To a certain extent, Deligne's exceptional series $A_1 \subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8$ plays a role analogous to the classical series $A_n \subset ...
7
votes
1answer
163 views
Simple Lie algebras: making subspaces 'very transversal'
Let $G$ be a Lie group or group of Lie type whose Lie algebra $\mathfrak{g}$ is simple. Because the Lie algebra is simple, for any proper subspace $V\subset \mathfrak{g}$,
there is a $g\in G$ such ...
3
votes
1answer
178 views
tensor product of massless Poincare representations
Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless ...
8
votes
1answer
123 views
Central extensions of loop groups
Let $LG=\operatorname{Maps}(S^1,G)$ be the loop group of a compact Lie group $G$. I should add some adjectives to $G$, but for sake of simplicity let's just take $G=SU(2)$.
There is a central ...
16
votes
1answer
275 views
Milnor Conjecture on Lie groups for Morava K-theory
A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the ...
3
votes
0answers
103 views
Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
9
votes
0answers
90 views
Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$
I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...
6
votes
0answers
94 views
Bordism groups and a short exact sequence
Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...
3
votes
0answers
83 views
Normalizer of a split torus
Let $G$ be a connected reductive group split over a field $k$. Let $T$ be a maximal split torus of $G$. Consider $N_G(T)$, the normalizer of $T$ in $G$, we have $N_G(T)/T \cong W$, the Weyl group of $...
16
votes
0answers
148 views
Limiting representation theory of quantum groups at roots of unity and SL(2,C)
Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
4
votes
1answer
145 views
The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators
Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$).
How is the embedding $\mathfrak{g}...
4
votes
4answers
350 views
Motivation for construction of Associated fiber bundle from a principal bundle
Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fibre bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)...
6
votes
1answer
155 views
Normal Fuchsian subgroups
I've been working with Fuchsian groups and from geometrical motivations finding a cocompact normal Fuchsian subgroups of $PSL(2,\mathbb{R})$ would have intresting properties for my research.
It is ...
3
votes
0answers
69 views
Induced $(\mathfrak{g},K)$-modules
Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
12
votes
3answers
614 views
Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...
3
votes
1answer
87 views
Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$
Let $F_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}_4$, with Dynkin diagram
$$
\beta_1-\beta_2\Rightarrow\beta_3-\beta_4\,.
$$
Let $P_{\{\beta_2\}}...
3
votes
0answers
54 views
Restriction that contains a trivial representation
Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$ such that $(G,G')$ is a symmetric pair. If $\pi$ is an infinitely dimensional unitary representation of $G$, ...
3
votes
0answers
96 views
Lie algebra of a compact Lie group
Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...
3
votes
1answer
120 views
Coinduced modules in the BGG category $\mathcal O$ over complex semisimple Lie algebras
For a given finite-dimensional complex semisimple Lie algebera $\mathfrak g$, we fix Cartan $\mathfrak h$ and Borel subalgebras $\mathfrak b$, then we have the BGG category $\mathcal O$. As usual, we ...
1
vote
0answers
125 views
Idempotents in Group Algebras
What is known about idempotents in Lie group algebras (such as on the classical Lie groups)? Specifically the self-adjoint ones. Is there anything interesting to say? I haven't been able to find much ...
3
votes
1answer
72 views
A partition of the set of order 2 outer automorphisms of $SU(N)$
Let $N$ be an even integer, $N>2$. Let $E$ be the set of all outer automorphisms $\phi$ of $G = SU(N)$ which are of order 2, i.e. $\phi \circ \phi = \mathrm{id}_G$.
Choose a particular element $\...
6
votes
0answers
165 views
Quotient space, a fundamental group, and higher homotopy groups 2
Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...
4
votes
0answers
234 views
Non-Abelian fundamental group? — a bizarre example
For the quotient space $G=G_0/G_1$, knowing the homotopy
groups of $G_0$ and $G_1$, one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(G_1) \to \pi_n(G_0) \to \...
2
votes
0answers
50 views
Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?
$\newcommand{\GL}{\operatorname{GL}}$
Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...
2
votes
0answers
197 views
Quotient space, homogeneous space, and higher homotopy groups
Preparation and my input:
For the quotient space $G/H$, knowing the homotopy
groups of $G$ and $H$ one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(H) \to \pi_n(G) ...
4
votes
0answers
118 views
Cyclic vectors for regular representations
I'm looking for references about the following aspect of cyclic vectors for regular representations.
Let $K$ be a compact Lie group. Let $K$ act on $L^2(K)$ by the left regular representation. Then $...
6
votes
1answer
203 views
Irreducible representation of the product of two groups and tensor product
Let $G_1, G_2$ be two lie groups, $V$ be a finite dimensional (continuous) irreducible complex representation of $G_1 \times G_2$, must $V \cong V_1 \otimes V_2$ for some irreducible representation $...
5
votes
0answers
43 views
Injectivity of exponential chart in a homogeneous space
Let $G$ be a finite-dimensional connected real Lie group and $\mathfrak{g}$ its Lie algebra. Let $\exp : \mathfrak{g} \to G$ denote the exponential map. Among the results proved independently by ...
10
votes
0answers
274 views
How to define (and compute) the Cartan-Killing form of the group of volume-preserving diffeomorphisms?
This question was raised a while ago in a blog post by Terry Tao on the Euler-Arnold equation and he called it "quite tricky".
Has anyone in the meantime tried to formulate this question precisely, ...
6
votes
1answer
105 views
Existence of a real eigenvalue is a necessary condition for the density of all the orbits of a Lie subgroup of $GL(\mathbb{R},d)$
Good morning,
I would like to pose the following (maybe naive) question. Let $\mathfrak{a}\subset \mathfrak{gl}(\mathbb{R},d)$ be any lie subalgebra, and $A$ be the connected, simply connected ...
7
votes
1answer
128 views
existence of riemannian metric on $\text{SL}_3(\mathbb{R})$ with special geodesics
Is there a left-invariant Riemannian metric on $\text{SL}_3(\mathbb{R})$ for which the geodesics (with respect to the corresponding Levi-Civita connection) through the identity are exactly the ...
22
votes
6answers
1k views
About the definition of E8, and Rosenfeld's “Geometry of Lie groups”
I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that ...
3
votes
1answer
158 views
Closed Semi-Riemannian manifolds with non-compact isometry group
Does anyone know a good reference for general results about closed Semi-Riemannian manifolds which have a non-compact isometry group?
Edit: My goal is to understand a bit better what the intuition ...
3
votes
0answers
243 views
Differential geometry and category theory
Does anyone know of a book/paper/anything, the longer the better introducing differential geometry from a category theoretic point of view? Everywhere it seems categorical language is the elephant in ...
4
votes
2answers
133 views
Generating Irreducible representations of a simple lie algebra with Schur functors
Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $...
3
votes
1answer
133 views
Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices
Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that
$$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$
But suppose I ...
4
votes
1answer
67 views
A question about centralizer of a vectors in the positive Weyl chamber
Given a compact Lie group $K$ and a maximal torus $T\leq K$, and choose a positive Weyl chamber $\mathfrak t^*_+\subset\mathfrak k^*$, where we used a $K$-invariant inner product on $\mathfrak k$. ...
5
votes
1answer
134 views
Can I bring the Kirillov 2-form on coadjoint orbits to adjoint orbits?
I tried asking this question on stackexchange and received no response.
Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I ...
5
votes
0answers
93 views
Convolution theorem on a non-abelian Lie group
Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the ...
