All Questions
Tagged with linear-algebra lie-algebras
91 questions
9
votes
1
answer
158
views
Eigenfunctions of the Laplace–Beltrami operator on the coadjoint orbit of $\mathfrak{su}(n)$
$\DeclareMathOperator\SU{SU}$For $\mathfrak{su}(2,\mathbb{C})$, the generic coadjoint orbit is $\mathbb{S}^2$, and the Laplace–Beltrami operator on it is given by
$$
\Delta \equiv \frac{1}{\sin\theta} ...
- 195
4
votes
0
answers
103
views
Relationship between characteristic polynomials of a matrix and its adjoint representation
Let $A \in \mathrm{M}_n(F)$ be a matrix over a field $F$. Consider its adjoint representation $\mathrm{ad}_A \in \mathrm{End}(\mathrm{M}_n(F))$, defined by
$$
\mathrm{ad}_A(X) = [A, X] = AX - XA.
$$
I ...
- 309
3
votes
1
answer
111
views
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I found myself needing a generalization of a result of Kostant in his famous paper
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math, Volume 34, 1979, ...
- 173
5
votes
0
answers
231
views
Avoiding Cartan subalgebra in a Lie algebra
Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation.
What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ ...
- 309
15
votes
1
answer
518
views
Pairs of matrices for which traces of powers are independent of the order
Let $A,B$ be $n\times n$ matrices over ${\mathbb C}$ such that, for all $m,k$ and all partitions $(i_1,\ldots ,i_r)$ of $m$ and $(j_1,\ldots ,j_r)$ of $k$ (perhaps with some zero parts),
$${\rm tr}\, (...
- 1,336
1
vote
0
answers
58
views
Linear algebraic group, absolute root system, computing roots
Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
- 11
1
vote
0
answers
139
views
Integral convex polytopes formed from the weight diagrams of representations of $\mathfrak {sl}_4$($\mathbb{C}$)
I'm a student studying undergraduate abstract algebra and doing a summer research project in the mathematics department at my school. I'm barely familiar with the rudiments of representation theory; I ...
- 111
6
votes
0
answers
340
views
Asymptotically nilpotent Lie sets of matrices
A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$.
Question 1. Assume that $\mathcal{A}$ is the subset of $\textbf{...
- 291
5
votes
1
answer
398
views
Asymptotically nilpotent matrices
A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$. Assume that $\mathcal{A}, \mathcal{B}$ be maximal (under ...
- 291
1
vote
1
answer
298
views
Degenerate representation
Let $\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$ be a faithful representation of the commutative Lie algebra $\mathbb{R}^n$ into the Lie algebra of skew-symmetric matrices. There is an orthonormal basis ...
- 442
1
vote
1
answer
348
views
General strategy of error bound of matrix exponential
I want to ask General strategy of the error bound of the matrix exponential.
For example, suppose, $A, B$ are finite dimension $n \times n$ matrices with complex coefficients. Using Baker–Campbell–...
- 149
3
votes
1
answer
69
views
Action of Coxeter element on mod $2$ root lattice is semisimple
Let $\Lambda$ be a simply laced root lattice and $w$ a Coxeter element of the Weyl group of $\Lambda$.
Question: Is it true that the action of $w$ on the $\mathbb{F}_2$-vector space $\Lambda/2\Lambda$ ...
- 984
2
votes
1
answer
168
views
Irreducible $G$-representations with unital algebra structure
Let us work over $\mathbb C$. Suppose that $G$ is a semisimple algebraic group and let $H \subset G$ be a maximal torus. Consider a dominant weight $\omega$, then one can associate a unique ...
- 381
1
vote
0
answers
71
views
What subspace of $\operatorname{SU}(4)$ group keeps an element of the $\mathfrak{su}(2)$ subalgebra within $\mathfrak{su}(2)$ upon adjoint action?
Consider the Lie group $G_4=\operatorname{SU}(4)$ with (15) generators $T^a$. A basis for the latter is
$$\{\sigma^j \times 1_2, \quad \quad \sigma^i \times \sigma^j, \quad \quad 1_2 \times \sigma^j\},...
- 155
1
vote
0
answers
143
views
Why is this operator independent of the choice of basis
I asked this question in MSE but I received no answer
https://math.stackexchange.com/questions/4009524/why-is-the-following-operator-independent-of-the-choice-of-basis/4013636#4013636
Let $G$ be a lie ...
- 139
4
votes
1
answer
199
views
Is every $M\in \mathfrak{s}\mathfrak{p}_4(F)$ conjugate to an "upper triangular" matrix?
Let $F$ be a field and write $$\mathfrak{s}\mathfrak{p}_4(F)=\left\{\left(\begin{array}{cc} A & B \\ C & -A^T \\ \end{array}\right)\mid A,B,C\in M_2(F), B=B^T, C=C^T\right\}$$ for the ...
- 141
16
votes
0
answers
755
views
Is there a "natural" proof of the equality $4^2=2^4$?
This question, or rather any answer that it might receive, would probably belong to the realm of Awfully sophisticated proof for simple facts. Still, I claim that I have quite serious motivation for ...
- 18.4k
32
votes
2
answers
1k
views
A question about subspace in ${\bigwedge}^2({\mathbb R}^n)$
Let $E$ be a linear subspace of ${\bigwedge}^2({\mathbb R}^n)$. What is the minimal dimension of $E$ that guarantees $E$ contains a nonzero element of the form $X\wedge Y$, with $X, Y\in{\mathbb R}^n$?...
- 637
11
votes
3
answers
587
views
Is every $A \in \mathrm{SL}_n(\mathbb C)$ a product of four unipotent matrices?
Is every matrix $A \in \mathrm{SL}_n(\mathbb C)$ a product of four unipotent matrices?
I have verified that this is true if $n = 2$, and I believe I have came across this result before. However, I ...
- 485
8
votes
2
answers
636
views
Bilinear forms in compact/semisimple Lie group theory
If you look up the list of compact or semisimple Lie groups, you will see that three out of four infinite families (B, C and D) are defined in terms of a bilinear form on a vector space, either ...
- 481
3
votes
0
answers
137
views
Generic linear subspaces of symmetric matrices
Let $\mathcal{S}_{n}(\mathbb{R})$ be the real vector space of symmetric $n\times n$ traceless matrices with real entries and let $L\subset \mathcal{S}_{n}(\mathbb{R})$ be a linear subspace. Noticing ...
- 3,020
1
vote
0
answers
96
views
Constructing homeomorphisms from continuous functions and matrix exponentials
Fix a $d\times d$ matrix $A$, let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous function, and define the induced map $F_{f,A}:\mathbb{R}^d\rightarrow \mathbb{R}^d$ by
$$
x \mapsto \exp(f(x)A)...
- 5,405
2
votes
1
answer
270
views
Explicit Normalizer of SU(3) Cartan subalgebra
The normalizer $N(\mathfrak{h})$ of the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{su}(3)$ is defined as
$$N=\left\{ x \in SU(3)\;|\; x^\dagger\mathfrak{h}x \in \mathfrak{h}\right\}$$
I would like ...
- 155
3
votes
1
answer
132
views
Linear Lie algebra generated by $\mathbb{R}$-diagonalizable matrices
If $\mathfrak{gl}_n(\mathbb{R})$ denotes the Lie algebra of real $(n \times n)$-matrices, then the $\mathbb{R}$-diagonalizable matrices generate $\mathfrak{gl}_n(\mathbb{R})$ as a Lie algebra.
If $\...
- 97
6
votes
0
answers
192
views
Bar notation in Bourbaki’s *Lie groups*, Chap. IX
I am looking at Chapter IX (Compact Real Lie Groups), §4, Exercise 8 (translation). Given a complex subspace $\mathfrak p$ in the complexification $\mathfrak g_{\mathbf C}$ of some $\mathfrak g$, they ...
- 31.5k
2
votes
1
answer
120
views
Can Hom-Lie algebras be seen as an $\Omega$-algebras?
An $\Omega$-algebra over a field $K$ is a $K$-algebra $A$ with a set of multilinear operators $\Omega$, where $\Omega=\bigcup_{m=1}^{\infty} \Omega_{m}$ and each $\Omega_{m}$ is a set of $m$-array ...
- 367
3
votes
1
answer
255
views
How does the constancy of an operator’s eigenvalues imply the integrability of its eigenvector distribution?
I am reading a paper, Yano and Ishihara, “Submanifolds with Parallel Mean Curvature Vector” (MSN), where the authors have constructed a linear operator, say $A$, on vector fields. They claim that ...
- 161
4
votes
2
answers
360
views
Double centralizer in special linear algebra
It is well known that for a matrix $A$ in $\mathfrak{sl}_n(\mathbb{C})$, we have the following equivalence:
$$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$
where $Z(A)$ is the ...
- 617
6
votes
2
answers
831
views
Cyclic vectors in irreducible representations of simple Lie algebras
Is there a notion of "cyclic element" in a simple Lie algebra? In particular, is it independent of the irreducible representation chosen?
Explanation.
An endomorphism A is called cyclic if there is ...
- 617
2
votes
0
answers
170
views
Infinitesimal matrix rotation towards orthogonality
TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal.
Setting
In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...
- 181
10
votes
0
answers
420
views
Gram matrix determinant in dimension 4 and $E_8$
Consider a determinant of a Gram matrix in dimension $4$.
$$\begin{vmatrix}
1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\
-\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\...
- 4,316
4
votes
0
answers
391
views
Trace of the adjoint action is an eigenvalue in $\mathrm{U}(L)$?
Let $L$ be a finite-dimentional complex Lie algebra. $\forall x \in L$, one defines the adjoint action of $x$ on $L$ as the map
$\mathrm{ad}_x : L \to L, \text{ with } \mathrm{ad}_x(y) = [x,y]$
for ...
- 71
2
votes
1
answer
207
views
Question on the proof of any finite dimensional module of a semisimple Lie algebra is semisimple
I'm going through a proof of the theorem, any finite dimensional module of a semisimple Lie algebra is semisimple, from page 10 of these notes (pdf). I am having a difficult time understanding few ...
- 3,625
8
votes
1
answer
253
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 ...
- 20.2k
8
votes
1
answer
799
views
higher Casimirs for $\mathfrak{sl}$
The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
- 8,597
2
votes
1
answer
924
views
Number of Symmetric matrices of fix rank over finite fields
This might be a question that shouldn't be asked here. But I need some help.
I want to count the number of $n\times n$ symmetric matrices over the finite field $\mathbb{F}_q$ and rank $r$. I found the ...
- 179
1
vote
0
answers
87
views
Maximum Number of Skew-Symmetric matrices
I want to count the maximum number of rank 2 matrices in a space of certain dimension but I am stuck at some point. Any help/ suggestions are appreciated. Here is the question.
Let $\mathbb{M}_m$ be ...
- 179
3
votes
0
answers
91
views
Number of vectors such that the projection is decomposable
Let $V$ be a vector space of dimension $n\geq 6$ over the finite field $\mathbb{F}_q$. Let $\omega\in\bigwedge^{n-3}V$ be a nonzero element. Define the annihilator subspace of $\omega$ by $V_\omega=\{...
- 179
0
votes
1
answer
390
views
An upper bound for skew symmetric rank 2 matrices
Earlier, I had asked a similar question but that was not the correct problem where I got stuck. After a few quick answer, I realized that and I apologize for that.
Let $B_m$ be the space of all skew-...
- 179
2
votes
0
answers
79
views
Conditions on a $n\times n$ Hermitian matrix such that its extremal eigenvectors have equal magnitude entries
Is it possible to find (necessary and sufficient) conditions on a general $n\times n$ Hermitian matrix $A$, such that its extremal eigenvectors (the eigenvectors corresponding to the maximum and ...
- 31
5
votes
1
answer
158
views
$SL_2$-action on the free lie algebra on a 2-dimensional vector space
Let $V$ be a 2-dimensional vector space (over, say, $\mathbb{Q}$). Let $FL$ be the free lie algebra on $V$, then there is a natural action of the group $SL(V)$ on $FL$, such that the action of $-I$ is ...
- 10.7k
2
votes
1
answer
2k
views
Parametrization of SL(3,R)
Are there any known common parametrizations of SL(3,R)? I know that it is easy to obtain a local parametrization by just exponentiating generators from the Lie algebra, but I do not know if they are ...
- 293
6
votes
1
answer
882
views
A question on the smallest singular value
Let $X(r)$ be the set of matrices $A \in M(n \times m)$, $n \leq m$, such that the norm of $A$ (largest singular value) is smaller or equal than $1$ and the smallest singular value of $A$ is smaller ...
- 61
1
vote
0
answers
460
views
Orthonormal basis of matrices
I am asking if somebody knows how to do or is aware of the following construction:
Let $n \in \mathbb{N}$ be given, then take an arbitrary matrix $A \in \mathbb{C}^{n \times n}$. Then, there are maps ...
- 21
1
vote
0
answers
398
views
Center of matrices
I encountered a neat problem in a problem in particle physics
So given $n$ skew symmetric matrices $A_1,...,A_n$ in $\mathbb{C}^{d \times d}.$
I would like to call this the commutator property: $...
1
vote
1
answer
358
views
Properties of semisimple Lie algebra elements
I'm looking for a source of properties for semisimple Lie algebra elements, specifically finite dimensional classical Lie algebras.
I start with the assumption that I have a complexified Lie algebra $...
- 23
1
vote
0
answers
173
views
Generating $\mathfrak{so}(7)$
Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is actually a subalgebra isomorphic to $\...
- 369
0
votes
0
answers
111
views
Sandwich rule for Lie algebras
On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...
- 1,959
2
votes
1
answer
201
views
multiplicity-free action on $SO(n+1)/SO(n-1)$
I'm trying to show that the Lie group $G=SO(n+1) \times SO(2)$ acts multiplicity-free on the cotangentbundle $T^* (SO(n+1)/SO(n-1))$.
That means:
1)
There exists an $\operatorname{Ad}^*_G$-...
- 501
2
votes
1
answer
367
views
Commuting nilpotent matrix collection
For every large enough $m\in\Bbb N$ are there $c=\alpha m$ (for some fixed $\alpha>0$) square matrices $A_1,\dots,A_c$ that commute with each other with nonzero product ($\forall i,j\in\{1,\dots,t\}...
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