Questions tagged [lie-groups]

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

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Tits construction of algebraic groups of type D₆ and E₇ via C₃

As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
nxir's user avatar
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Literature on Lyndon words and the Lie commutator

Since I lost my paper notes in a domestic conflagration in Japan some ten years ago, I've occasionally tried to recall one particular author who wrote in the 1900s about Lyndon words / strings, or ...
Tom Copeland's user avatar
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Deformations of the 4-sphere with 8-dimensional isometry groups

I am looking for deformations of the 4-sphere with 8-dimensional isometry group, like a 4-dimensional Berger sphere.
Thomas Schucker's user avatar
6 votes
1 answer
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Is there a general method for computing finitely generated normalizers?

I'm looking to compute normalizers of finite subgroups of $\mathrm{GL}(n, \mathbb{Z})$ and its possible that they are infinite but they are always finitely presented. For $\mathrm{GL}(n, \mathbb{Z})$ ...
Jim's user avatar
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Step in the Bruhat decomposition for reductive Lie groups

Err, not research but if anyone has read this part of Knapp's book recently, I'd be obliged if they could help me out. Also posted on MSE. I'm stuck on a line in the proof of Theorem 7.40 in Knapp's '...
Chertopkhanov on Malek Adel's user avatar
2 votes
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Complex semisimple Lie algebra modules with non-semisimple Cartan action

Let $\frak{g}$ be a complex semisimple Lie algebra. I would like to know about infinite-dimensional representations $M$ of $\frak{g}$ for which the Cartan $\frak{h} \subseteq \frak{g}$ does not act ...
László Szabados's user avatar
4 votes
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Why are all "non-swinging" representations self-dual?

Let $\mathfrak{g}$ be a semisimple (say complex) Lie algebra, and $V$ an irreducible finite-dimensional representation of $\mathfrak{g}$. Denote by $w_0$ the longest element of the Weyl group, i.e. ...
Ilia Smilga's user avatar
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Another question about unitary and anti-unitary matrices

This question is related to the previous post, "A question about unitary and anti-unitary matrices". Following the suggestion of Lspice, I am posting it as a separate question, as it might ...
jacaboul's user avatar
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Non-uniform lattices and parabolic subgroups in Lie groups

Let $G$ be a semisimple connected Lie group and let $\Lambda < G$ be a non-uniform irreducible lattice. How does it follows that there exists a minimal parabolic subgroup $P$ of $G$ such that the ...
Constantin K's user avatar
7 votes
1 answer
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Infinite dimensional representations of $\frak{sl}_2$

The finite-dimensional representations of a complex semisimple Lie algebra $\frak{g}$ are well known to be classifiable by their highest weight vectors, giving a convenient countable indexing set. I ...
László Szabados's user avatar
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Methods for calculating (one-parameter subgroup) actions

For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form \begin{equation} \mathrm{e}^{t L(z)} f(z) \end{equation} ...
horropie's user avatar
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Non-integrable almost complex structure for complex projective $3$-space

It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this article for ...
Didier de Montblazon's user avatar
9 votes
2 answers
662 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
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Are there invariants of configurations of points in space obtainable via the moduli space of solutions of the Berry-Robbins problem?

Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in Euclidean $3$-space and let $U(n)/T^n$ denote the flag manifold associated to the unitary group $U(n)$, i.e. the ...
Malkoun's user avatar
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29 votes
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Gaussian elimination is just Gram-Schmidt with a change to the inner product symbol?

I noticed at some point that if you take the Gram-Schmidt algorithm for taking the QR decomposition of a matrix, and you change the meaning of the inner product symbol $\langle \mathbf u, \mathbf v \...
wlad's user avatar
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Representation determined by traces

A discrete, faithful representation of a surface group $G:\pi_1(S_g) \to PSL_2(\mathbb{R})$ is determined, up to conjugacy by $PGL_2(\mathbb R)$, among such representations by the squares of traces of ...
RegularGraph's user avatar
6 votes
1 answer
258 views

Peterson's quantum cohomology of G/P lectures

Dale Peterson famously gave a series of lectures on the quantum cohomology of flag varieties $G/P$ at MIT in 1997. These lectures are often cited in subsequent papers by other authors on the subject (...
SamJeralds's user avatar
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Calculation of first correction to Selberg type integral

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\arcsinh{arcsinh}$Let $U \in G$, where $G$ is $\SU(N)$ matrix. $\Tr U$ will denote the character ...
Sergii Voloshyn's user avatar
0 votes
1 answer
117 views

Sub-coroot lattices

[This is a sequel to the previous question sub-coroot systems, that has been answered! :-) ] Let $T$ be a maximal torus of a compact Lie group $K$, and let $\Lambda \subset {\mathfrak t}$ be the ...
bernardorim's user avatar
3 votes
1 answer
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Sub-coroot systems

Let $T$ be a maximal torus of a compact Lie group $K$, and let $\Psi \subset {\mathfrak t}$ be the (finite) set of coroots for $(K,T)$, where $\mathfrak t$ is the Lie algebra of $T$. Assume now that $...
bernardorim's user avatar
7 votes
1 answer
265 views

Non-homogeneous line bundles over a homogeneous space

Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A homogeneous vector bundle over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form $$ G \times_{\...
László Szabados's user avatar
3 votes
1 answer
250 views

Decomposition of tensor powers of the vector representation of $\frak{sl}_n$

Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as $$ V(\pi_1) \...
László Szabados's user avatar
2 votes
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Multiplicative invariants of non-reduced root systems

It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...
G. Gallego's user avatar
2 votes
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126 views

Visualisation of general 3x3 matrices, with applications to the pedagogy of linear algebra?

I've got a method for visualising non-zero $2 \times 2$ real matrices (modulo non-zero scalar factor) using the fact that: Nonnegative determinant matrices (modulo non-zero scalar factor) are in 1-to-...
wlad's user avatar
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1 vote
1 answer
187 views

Tensoring irreducible representations corresponding to root lattice elements

Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible ...
Martim Pereir's user avatar
0 votes
1 answer
109 views

Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules

Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial? ...
Martim Pereir's user avatar
4 votes
1 answer
232 views

Number of representations of a semisimple Lie algebra of any given dimension

For a semisimple complex Lie algebra $\frak{g}$ it is well known that irreducible finite-dimensional representation are not characterised by their dimension. More formally, let us define an ...
Martim Pereir's user avatar
3 votes
1 answer
361 views

Topological vector spaces in direct sum

A year ago, I asked this question here at Mathematics Stackexchange, but no one there managed to answer it. So I am elevating it to MathOverflow. This question had emerged as an offshoot of a bigger ...
Michael_1812's user avatar
2 votes
1 answer
213 views

What do the Pauli matrices say about the Threefold Way?

The Pauli matrices $$\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1 \...
Andrius Kulikauskas's user avatar
1 vote
1 answer
249 views

Problem in understanding the coadjoint action of $\mathfrak {g}^{\ast}$ on $\mathfrak {g}$

$\DeclareMathOperator\ad{ad}$Let $\mathfrak {g}$ be a Lie bialgebra. Then $\mathfrak {g}^{\ast}$ is also a Lie bialgebra which is dual to $\mathfrak {g}$. Let the brackets on $\mathfrak {g}$ and $\...
Anil Bagchi.'s user avatar
1 vote
0 answers
88 views

A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori

$\newcommand{\Hom}{{\rm Hom}} \newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}} \newcommand{\X}{{\sf X}} $ I am looking for a reference for the following lemma (for which I know a proof): Lemma. Let $\...
Mikhail Borovoi's user avatar
4 votes
0 answers
133 views

Finite subgroups of $\mathrm{SL}_2(\mathbb{O})$

What are the finite subgroups (subloops that are groups) of $\mathrm{SL}_2(\mathbb{O})$? In particular, are there any not contained in a $\mathrm{SL}_2(\mathbb{H})$?
Daniel Sebald's user avatar
4 votes
1 answer
267 views

Is a Lie subgroup whose center is closed, a closed subgroup itself?

I want to show that a certain Lie subgroup (i.e. generated by the exponential of elements in some Lie subalgebra) of a Lie group is closed. My knowledge of the subject of Lie groups is rudimentary, ...
Pablo Lessa's user avatar
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0 votes
1 answer
164 views

Sum of weights of an irreducible representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
Blind Miner's user avatar
2 votes
1 answer
470 views

Classification of Lie group structures on $\mathbb{R}^n$

Is it possible to describe, up to isomorphism, all Lie groups $G$ whose underlying manifold is diffeomorphic to $\mathbb{R}^n$ (with its standard smooth structure)? In fact, I haven't found any such ...
Arshak Aivazian's user avatar
2 votes
2 answers
457 views

Complete representation theory of $\mathrm{SL}(2,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...
Arnold Neumaier's user avatar
2 votes
1 answer
88 views

Unitary dual of universal cover

The universal covering group $G$ of $\mathrm{SL}_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\...
user avatar
0 votes
0 answers
144 views

The largest abelian subgroups of a Lie group

Let $G$ be a semisimple Lie group. Denote $d(G)$ as the maximal integer $p$ such that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\...
Yushi MuGiwara's user avatar
3 votes
0 answers
138 views

Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$ G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) . $$ Basic ...
André Schlichting's user avatar
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0 answers
65 views

Why is $\Pi_r^L$ a non-degenerate Poisson structure on $G\ $?

Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...
Anil Bagchi.'s user avatar
4 votes
0 answers
160 views

The homotopy type of the space of symplectic structures

While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
Akerbeltz's user avatar
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0 answers
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Left translations respect the Schouten bracket

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and $r \in \bigwedge^2 \mathfrak{g}$. For $x \in G$ let $\lambda_x$ denote the left multiplication by $x$. Let $[\cdot, \cdot]$ ...
Anil Bagchi.'s user avatar
4 votes
1 answer
398 views

Faithful locally free circle actions on a torus must be free?

Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is free? I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$. Another related question is: ...
chan kifung's user avatar
4 votes
1 answer
254 views

Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made. In this paper, $\mathfrak{g}$ is a ...
Ji Woong Park's user avatar
3 votes
1 answer
128 views

Does every nilpotent orbit have an element supported on the simple root spaces?

Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha_1, \dots ,\...
user492133's user avatar
3 votes
0 answers
135 views

Summing over roots of a simple Lie algebra and Deligne series

For a simple Lie algebra $\mathfrak{g}$ we can define a Killing form $K(X,Y) \equiv \frac{1}{2 h^\vee}\operatorname{tr}(\mathfrak{ad}_X \mathfrak{ad}_Y)$, where $\mathfrak{ad}_X Y \equiv [X, Y]$ as ...
Lelouch's user avatar
  • 857
4 votes
0 answers
188 views

Almost conjugate subgroups of compact simple Lie groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group. Definition: Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...
emiliocba's user avatar
  • 2,321
1 vote
1 answer
176 views

A question from the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel

In the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel, the author has made the following statement in the proof of Corollary D in page no. 18 ( https://arxiv.org/pdf/...
John Depp's user avatar
  • 187
0 votes
1 answer
144 views

Centralizer of a reductive subgroup

Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...
Windi's user avatar
  • 833
1 vote
0 answers
60 views

Choice of generators to make the centralisers connected

In $G=\operatorname{PGL}_{2n}(\textbf{C})$, WLG, we assume all the toral elementary abelian 2-subgroups in discussion are in $T$, the image in $G$ of the group of diagonal matrices in $\operatorname{...
user488802's user avatar

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