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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

13
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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
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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
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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
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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
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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 ...
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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
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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
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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 ...
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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
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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
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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
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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
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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/...
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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
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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 ...