Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
13
votes
4answers
403 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 ...
2
votes
0answers
67 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
211 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 ...
3
votes
2answers
106 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
121 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
63 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
95 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
83 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 ...
1
vote
1answer
102 views
Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?
In Witten's paper Three Dimensional Gravity Revisited and Quantization of Chern-Simons Theory with Complex Gauge Group, he used a fact that for a principal $G$-bundle, the quantization of the Chern ...
7
votes
0answers
165 views
What is $SL(2,\mathbb{R})$-Chern-SImons Theory?
I found in physics that Chern-Simons theory is closely related with three dimensional gravity.
From this paper Three Dimensional Gravity Revisited, the author talks about the Chern-Simons for
$$\...
7
votes
0answers
44 views
Criterion for existence of a homogeneous space associated to a Lie pair
Recall that every finite-dimensional real Lie algebra $\mathfrak{g}$ is the Lie algebra of a simply-connected Lie group, which is unique up to isomorphism.
This statement generalises somewhat to ...
4
votes
1answer
62 views
Multiplicities in Plancherel theorem for SL2(R)
The usual formulation of the Plancherel theorem one writes $f(1)$ as an integral over the dual $\widehat G$. The support of the measure is the set of representations which weakly occur in $L^2(G)$. ...
0
votes
0answers
58 views
How the roots and weights changed under a folding?
Let $e_1,e_2,e_3,f_1,f_2,f_3$ be the generators of the Chevalley basis of the Lie algebra $sl_4$. Let $e_1' = e_1+e_3$, $e_2'=e_2$, $f_1'=f_1+f_3$, $f_2'=f_2$. Then the subalgebra generated by $e_1', ...
3
votes
1answer
197 views
Slice theorem for proper groupoids
Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$.
Assume $G$ is étale, i.e., the source and range maps of $G$ are local ...
-2
votes
0answers
138 views
What are the universal covers of $G_2$ as a Lie group?
What is the universal cover of $G_2$ as Lie groups? In particular, how do we express $G_2$ as the quotient of universal covers by deck transformations $\tilde{G}/Aut(G)$?
Also, this is perhaps a ...
1
vote
0answers
60 views
Volume form preserved by the action of $\mathrm{PGL(n+1, \mathbb{R}})$ on $\mathbf{P}^n(\mathbb{C}) \setminus \mathbf{P}^n(\mathbb{R}) $?
I know this is quite an elementary question but I am not an expert in Lie theory.
Does the action of $\mathrm{PGL(n+1, \mathbb{R}})$ on $\mathbf{P}^n(\mathbb{C}) \setminus \mathbf{P}^n(\mathbb{R}) $ ...
1
vote
0answers
122 views
Is there a class of non-symmetric spaces $G/SO(n)$ such that $Hol(G/SO(n))=SO(n)$?
Is there a certain class (or unique construction of a set) of simply connected non-symmetric spaces $G/SO(n)$ such that the Riemannian holonomy is $Hol(G/SO(n))=SO(n)$? (That is, is there a way to ...
1
vote
0answers
50 views
sequence definition of proper group action
My understanding is that for an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the map $G \times M \rightarrow M \...
2
votes
0answers
75 views
Quotient by a non-free action of a Lie group and manifolds with corners
The quotient manifold theorem says that
If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/G$...
4
votes
0answers
68 views
Representation on square integrable sections of a principal bundle
Let $X\rightarrow Y$ be a smooth principal $G$-bundle for some Lie group $G$. Then $L^2(X)$ has a natural $G$-action determined by fibrewise action of $G$ on $X$.
We have an abstract isomorphism of ...
5
votes
1answer
186 views
Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced?
A subset $B$ of a group $G$ is called balanced if $gBg^{-1}=B$ for all $g\in G$.
An action of a group $G$ on a metric space $X$ is called ballanced if for each non-empty balanced subset $B\subset G$ ...
2
votes
0answers
55 views
Lie algebra bundle associated to a Lie group bundle
I was reading something(page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google gave http://www.pphmj.com/Images/...
2
votes
0answers
69 views
What are the simply-connected non-compact irreducible symmetric spaces? [migrated]
Would someone be able to list (or provide a reference to) the simply-connected non-compact irreducible symmetric spaces of rank $\ge 1$(as quotients of Lie groups $G/H$)?
Any help would be ...
5
votes
1answer
192 views
Fell's trick for Lie groups
Let $\Gamma$ be a countable discrete group and $\lambda$ the left regular representation of $\Gamma$ on $l^2(\Gamma)$. Let $\rho:\Gamma\rightarrow U(H)$ be a unitary representation of $\Gamma$ on some ...
5
votes
0answers
76 views
Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
6
votes
0answers
156 views
Relation between Lie group characters and spherical functions on symmetric spaces
Setup: Let $G/K$ be an irreducible compact Riemannian symmetric space, where $G$ is a simply connected compact real Lie group, and $K$ is the maximal compact connected subgroup of some noncompact real ...
5
votes
0answers
121 views
Restriction of discrete series
QUESTION
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals the rank of its maximal compact subgroup. Suppose that $G'$ is a reductive subgroup of $G$ with equal rank. If $\...
3
votes
1answer
173 views
Left invariant connections on a Lie group
The exponential map associated with the (-) - connection on a Lie group is generally not surjective. This is because, for this connection, the one-parameter subgroups and geodesics coincide. If we ...
3
votes
1answer
125 views
Holonomy groups of compact Riemannian symmetric spaces
Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:
https://en.wikipedia.org/wiki/...
-2
votes
1answer
103 views
Prove or disprove Any subgroup of differentiable lie group is analytic submanifold? [closed]
I have made a research in the web to know more about analyticity of of lie group and to give a convinced answer to this question which i have accrossed it in my research which is stated the following :...
4
votes
2answers
224 views
Weyl's Branching Rule for $SU(N)$-Setting
On the Wikipedia page for restricted representations
https://en.wikipedia.org/wiki/Restricted_representation
there is presented a number of explicit "branching rules". In particular, there is the ...
4
votes
0answers
67 views
Good range and fair range
Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
3
votes
0answers
109 views
Parallel transport for variety over finite field
I was wondering: Given a variety over a finite field, say the projective plane or sphere over $\mathbb{F}_q$. Then I can try to define parallel transport along (geodesic) curves. In particular, I can ...
4
votes
0answers
66 views
Modular $S$-matrix for an extended affine Lie algebra
This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.
In this paper, the authors ...
2
votes
0answers
55 views
Effective actions by non-commutative groups have non-commuting fundamental vector fields?
I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :)
Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ ...
2
votes
1answer
84 views
Descending chain property for compact Lie goups
I'm searching for a good reference that prove the descending chain propriety for compact Lie groups (i.e. every sequence $K_1\supset K_2\supset...$ of closed subgroups $K_i$ of $G$ is eventually ...
4
votes
1answer
87 views
Haar unitaries with constraints
Given that one can sample unitaries from the Haar measure over $U(n)$ (as in F. Mezzadri, Notices of the AMS 54 (2007), 592-604), how can one sample from the uniform distribution over the following ...
8
votes
1answer
197 views
Young tableaux for exceptional Lie algebras
Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does ...
1
vote
0answers
66 views
Modular transformation of affine characters of non-simply connected groups$.$
Consider an (untwisted) affine algebra corresponding to a compact and simply-connected Lie group $G$. Under a modular transformation, its characters transform as (cf. 9612078)
$$
\chi_\mu\to\sum_{\nu\...
6
votes
1answer
173 views
Branching from $E(6)$ to $SO(10) \times U(1)$
In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups
$$
SO(10) \times U(1) \hookrightarrow E_6
$$
is important object of interest. See here for my motivating example.
In ...
3
votes
0answers
40 views
Decay of Fourier coefficients for Hölder functions on compact Lie groups
If $f$ is a complex-valued function on a compact Lie group $G$, we have a decomposition $f = \sum_\mu f_\mu$ corresponding to the Peter-Weyl decomposition $L^2(G) = \oplus_\mu (\dim \mu) V_\mu$.
For $...
1
vote
1answer
111 views
Abstracting the properties of the category $\frak{g}$-modules
Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^*$, we know that the decomposition of $V \otimes V^*$ ...
11
votes
1answer
312 views
Decategorification of Gaitsgory's strange functional equation?
Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have ...
3
votes
1answer
54 views
The role of semisimple factors in transitive group actions on manifolds
I am trying to remember a result stating that under certain assumptions,
given a transitive smooth action of a (compact?) Lie group on a smooth manifold, also the action of the semisimple factor is ...
3
votes
1answer
200 views
Possible groups appearing in a Shimura datum
Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for ...
6
votes
1answer
201 views
Closures of torus orbits in flag varieties
Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$.
Now, as far as I ...
4
votes
0answers
42 views
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 ...
3
votes
0answers
63 views
Generalizing Autonne-Takagi factorization
Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that:
A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
6
votes
1answer
257 views
The group of Isometries of Shahshahani metric
Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\ldots,n\}$$
For $X=(x_1,x_2,\ldots,x_n)\in M$ put $|X|=\sum_{i=1}^n x_i$.
We consider the Shahshahani Riemannian metric $g$ on ...
2
votes
0answers
79 views
Fixed point set with non-empty interior
Let $G$ be an infinite compact separable Hausdorff metric group, and $H\subset G$ a closed subgroup, such that the left $G$-action on $G/H$ is effective (i.e., $H$ doesn't contain a non-trivial closed ...
