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59 votes
4 answers
15k views

Group theory in machine learning

I'm a Machine Learning researcher who would like to research applications of group theory in ML. There is a term "Partially Observed Groups" in machine learning theory which has been ...
drosophyllum's user avatar
33 votes
8 answers
9k views

"Modern" proof for the Baker-Campbell-Hausdorff formula

Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula? All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and are not at all geometric (...
Mark.Neuhaus's user avatar
  • 2,074
24 votes
3 answers
2k views

Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group

There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try ...
Gro-Tsen's user avatar
  • 32.5k
23 votes
1 answer
1k views

Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups. Below is a proposition for ...
Joonas Ilmavirta's user avatar
21 votes
2 answers
1k views

Geometric interpretation of exceptional symmetric spaces

Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...
JME's user avatar
  • 3,022
20 votes
6 answers
4k views

Polynomial invariants of the exceptional Weyl groups

Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
Christopher Drupieski's user avatar
20 votes
4 answers
3k views

Online References for Cartan Geometry

I would like to learn more about Cartan Geometry ("les espaces généralisés de Cartan"). I ordered Rick Sharpe's book "Differential Geometry: Cartan's generalization...", which would take a long time ...
Malkoun's user avatar
  • 5,215
20 votes
2 answers
1k views

Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative

I saw this problem some years ago and I would greatly appreciate any reference or solution. Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} )$...
jack's user avatar
  • 3,153
19 votes
1 answer
1k views

A result on Lie group actions on 15-dimensional spheres?

In this interview by Eric Weinstein to Roger Penrose, Timestamp 1:24:05., what result is the host talking about? Transcription of the relevant part: "If you have two sets of symmetries, known as ...
Qfwfq's user avatar
  • 23.3k
19 votes
3 answers
1k views

Is there "Schur-Weyl duality" for infinite dimensional unitary group?

To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
Michał Oszmaniec's user avatar
18 votes
1 answer
631 views

Best texts on Lie groups for number theorists

What are the most comprehensive textbooks on the structure of Lie groups and their infinite-dimensional representations if one is interested in their applications to number theory (so covering ...
user163784's user avatar
17 votes
4 answers
5k views

Exceptional isomorphisms of Lie groups

It is known that in low dimensions certain exceptional isomorphisms arise between Lie groups. I have read about some of them in some papers, but I have not been able to find a "systematic" treatment ...
Vader's user avatar
  • 171
17 votes
2 answers
2k views

Where did Sophus Lie write the group commutator for two one parameter groups

If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group commutator of the ...
Peter Michor's user avatar
  • 25.3k
17 votes
1 answer
1k views

References for Langlands classification

I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group. My ...
Malkoun's user avatar
  • 5,215
17 votes
2 answers
1k views

What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?

Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
skupers's user avatar
  • 8,167
16 votes
5 answers
2k views

About the intrinsic definition of the Weyl group of complex semisimple Lie algebras

It may be a easy question for experts. The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$...
Zhaoting Wei's user avatar
  • 9,019
16 votes
0 answers
188 views

Representation theory of Pin groups

I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point. Proctor describes this rule for $m = 2$...
Ilia Smilga's user avatar
  • 1,574
15 votes
2 answers
2k views

Isomorphism between Spin(3,2) and Sp(4, R)

I've been using the fact that Spin(3,2) is isomorphic to Sp(4, R) for a while, but I've never seen a proof. Can anyone point me in the direction of a good reference?
MRD1729's user avatar
  • 393
15 votes
1 answer
612 views

Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?

Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...
Hang's user avatar
  • 2,789
15 votes
2 answers
613 views

Existence of a regular semisimple element over $\mathbb{F}_{q}$

This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help. Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{...
D. Dona's user avatar
  • 455
15 votes
1 answer
951 views

Duistermaat and Kolk's lost chapters on Lie groups

In Duistermaat and Kolk's book Lie Groups, it is written in the preface that "the text contains references to chapters belonging to a future volume". I could not find this second volume anywhere. Has ...
wer's user avatar
  • 151
14 votes
5 answers
1k views

History of the notion of $(G,X)$-structure

I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work. So far, it appears that he was the first to set it. Many mathematicans ...
R. Alexandre's user avatar
14 votes
1 answer
503 views

Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent?

I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group. Is it true? Where can I find a proof? A counterexample for open subsets of $...
Gian Maria Dall'Ara's user avatar
14 votes
1 answer
786 views

The fifth k-invariant of BSO(3)

From work of Pontryagin and Whitney, as I understand it, the homotopy 4-type of $BSO(3)$ is $K(\mathbb{Z}/2,2) \times_{K(\mathbb{Z}/4,4)} K(\mathbb{Z},4)$, where the pullback is along the maps $\...
David Roberts's user avatar
  • 35.5k
13 votes
4 answers
2k views

Groups of matrices in which all elements have all eigenvalues equal in modulus

I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...
Ian Morris's user avatar
  • 6,206
13 votes
3 answers
2k views

on the center of a Lie group

I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles... First some definitions: • Let $G$ be compact, simple, and simply ...
André Henriques's user avatar
13 votes
0 answers
475 views

Singular cohomology of $BG$ and Borel cohomology of $G$

Stasheff, in "Continuous Cohomology of Groups and Classifying Spaces", attributes the following result to Wigner. For $A$ a discrete abelian group and $G$ a finite dimensional locally compact, $\...
mme's user avatar
  • 9,580
12 votes
3 answers
849 views

$A_{\infty}$-structure on closed manifold

Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ? Edit: First, ...
Ilias A.'s user avatar
  • 1,974
12 votes
6 answers
4k views

Representation Theory of Lie Groups: Reference Request

I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their Lie algebras. More precisely, ...
user73423's user avatar
  • 131
12 votes
3 answers
2k views

What is a good introduction to branching rules in representation theory?

I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups. When a Lie group has a set of irreducible representations, I'd like to know ...
Manuel Bärenz's user avatar
12 votes
2 answers
849 views

Groups associated with infinite dimensional Lie algebras

There is a classical correspondence between Lie algebras (over $\mathbb{R}$ or $\mathbb{C}$) and Lie groups in the finite dimensional case: to every Lie group $G$ there is an associated Lie algebra $\...
shane.orourke's user avatar
12 votes
1 answer
684 views

Is every connected semisimple linear Lie group the connected component of (the real points of) an algebraic group?

Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group? I was told some fact along this line is true but could not find any ...
Jerry's user avatar
  • 511
11 votes
1 answer
1k views

Classification of (not necessarily connected) compact Lie groups

I am looking for a classification of compact (not necessarily connected) Lie groups. Clearly, all such groups are extensions of a finite "component group" $\pi_0(G)$ by a compact connected ...
Ben Heidenreich's user avatar
11 votes
1 answer
222 views

Reference requests: Integral cohomology of $G_2$-homogeneous spaces

Do you know a place where the integral cohomology of $G_2$-homogeneous spaces is computed? Great computational efforts using representation theory in order to determine the ...
Nicolas Boerger's user avatar
11 votes
1 answer
383 views

Is there a comprehensive survey of the discrete series representation of a real reductive group?

Vague form of the question: where can one find a comprehensive and possible modern account of the discrete series representations of a real reductive group? Motivation: I am a master's student trying ...
Daniel Miller's user avatar
10 votes
5 answers
3k views

Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...
user47245's user avatar
  • 101
10 votes
3 answers
2k views

nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
cleanplay's user avatar
  • 245
10 votes
2 answers
3k views

References on Lie groups and dynamical systems

I'm interested in Lie theory and its connections to dynamical systems theory. I am starting my studies and would like references to articles on the subject.
10 votes
2 answers
914 views

Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$

Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers? I can only find ...
Mare's user avatar
  • 26.5k
10 votes
1 answer
262 views

What is the hidden symmetry behind four generic planes in $\mathbb{R}^4$?

Consider the action of $\operatorname{GL}(\mathbb{R}^4)$ on the Grassmannian of 2-dimensional subspaces of $\mathbb{R}^4$. In experiments, I observe that four randomly drawn points in this space are ...
Dustin G. Mixon's user avatar
10 votes
3 answers
508 views

Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R). The idea is that each non-trivial ...
Didier Collard's user avatar
10 votes
2 answers
538 views

Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to {\...
Misha's user avatar
  • 31.2k
10 votes
1 answer
354 views

Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

On the last page of Schmid's article "Discrete Series", he says "In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb ...
Allen Knutson's user avatar
10 votes
0 answers
308 views

Compact Lie groups are rational homotopy equivalent to a product of spheres

According to [1] and [2], it is “well-known” that a compact Lie group $G$ has the same rational homology, and according to [2] is even rational homotopy equivalent, to the product $\mathbb{S}^{2m_1+1} ...
Gro-Tsen's user avatar
  • 32.5k
10 votes
0 answers
269 views

differentiating positive energy LG reps

Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...
André Henriques's user avatar
9 votes
1 answer
874 views

Proofs that the conformal group in dimension $\ge 3$ is a Lie group

Let $M$ be a smooth manifold of dimension $\ge 3$, equipped with a conformal structure (or a Riemannian metric). Then, the group of conformal diffeomorphisms is a finite dimensional Lie group. A ...
Asaf Shachar's user avatar
  • 6,741
9 votes
3 answers
894 views

Representation rings of exceptional Lie groups

Let $G$ be a compact Lie group and let $R(G)$ denote its complex representation ring. If $G$ is simply connected, such as $G_2$, $F_4$ or $E_8$, then it is known that $R(G)$ is a polynomial ring [F. ...
Rasmus's user avatar
  • 3,174
9 votes
3 answers
576 views

Reference Request: Structure constants for G2

Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
D_S's user avatar
  • 6,180
9 votes
1 answer
750 views

Learning from unsuccessful attempts at the Poincaré conjecture

This is reposted from MSE, but perhaps it is more appropriate to post it here. Let me know if I'm wrong. Among the many unsuccessful attempts at solving the Poincaré conjecture, I'm wondering if there ...
Paul Cusson's user avatar
  • 1,763
9 votes
1 answer
361 views

Homotopy groups $\pi_{4n-1}(SO(4n))$

There is a very natural way to define generators of $\pi_{4n-1}(SO(4n))\cong \mathbb{Z}\oplus \mathbb{Z}$ in terms of quaternions when $n=1$ and octonions when $n=2$ (see for example Tamura, On ...
Kafka91's user avatar
  • 641

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