-1
votes
1answer
126 views

Highest weights of irreducible components of tensor product of irreducible sl(3)-module [closed]

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
5
votes
1answer
279 views

History of Jordan Canonical Form?

Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in: When and how was it first stated? (I understand it was independently stated ...
0
votes
1answer
160 views

when $g^*$ is invariant under $Ad(G)$?

Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra. Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under ...
3
votes
1answer
136 views

Reduction of antisymmetric complex matrices

Let $E=\mathfrak{so}(n,\mathbb{C})$ be the Lie algebra of antisymmetric complex matrices. We consider the action of the complex orthogonal group $SO(n,\mathbb{C})$ on $E$ by conjugation. Is there a ...
0
votes
1answer
124 views

L a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L).

Let L be a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L). (Der is short for derivation). In order to show that ad(L)=Der(L), the proof I followed proves that that the ...
4
votes
2answers
267 views

Is the condition ``adjoint action does not have eigenvalue $-1$" dense in a Lie group?

I need to answer (affirmatively, I hope) the following question: In a Lie group $G$ whose Lie algebra $\mathfrak{g}$ is equipped with an $\mathrm{Ad}$-invariant scalar product, is the open subset ...
8
votes
1answer
459 views

ad (A^n) is a polynomial in ad A ?

Let $k$ be a field and $n$ a nonnegative integer. For any matrix $U\in\mathrm{M}_n\left(k\right)$, let $\mathrm{ad} U$ denote the map $\mathrm{M}_n\left(k\right)\to \mathrm{M}_n\left(k\right),\ ...
6
votes
2answers
820 views

Simple Lie algebras and Jordan decomposition

Let $F$ be an algebraically closed field and let $L$ be a simple Lie algebra of dimension $n$ over $F$. Let $ad: L\longrightarrow End_F(L)$ denote the adjoint representation of $L$. If $F$ has ...
2
votes
1answer
204 views

An innocent looking subgroup of $U(n)$

Consider the Lie subalgebra of $\mathfrak{u}(n)$ given by $L = \{A \in \mathfrak{u}(n): \sum_{j=1}^n A_{ij} = 0 \text{ for all } i \in [n]\}$. What is its dimension? What does the corresponding Lie ...
3
votes
2answers
984 views

Logarithm of a matrix

I am looking for a reference to study logarithm of an invertible triangular matrix. What is a good algorithm? Are there any good reference which studies this topic both theoretically and from an ...
2
votes
3answers
456 views

A question on the root systems of simple Lie algebras in the 90 degree case

I've been taking a look at simple Lie algebras for particle physics and I've found myself wondering about the following question. It can be shown that the adjacent roots in a root diagram ...
4
votes
1answer
528 views

Nilpotent matrices related to Lie algebras of special orthogonal groups in characteristic 0

In terms of matrix theory, the question I'm led to is the following: Start with an $n$-dimensional vector space over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, which has ...
4
votes
2answers
1k views

How many commuting nilpotent matrices are there?

To be precise, fix $n$, fix a field $k$. What is the maximal dimension of a subspace of the vector space of all $n\times n$ matrices formed by commutative nilpotent matrices? By commutative I mean ...
5
votes
1answer
206 views

Modeling free Lie algebras with matrix algebras

I am approximating some algebraic expressions of operators from a free Lie algebra. It is possible but messy to collect all independent operator objects of a given degree (same as grading?) that ...
10
votes
2answers
827 views

How are these two ways of thinking about the cross product related?

I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free ...
4
votes
2answers
489 views

Under what conditions will a unitary matrix fix a subspace which does not diagonalize the generating Hamiltonian?

Hello, this is my first post here. I hope that it is not too vague; I will be as precise as I can, but I have more of a meta-problem so please forgive me if this is inappropriate. My question is ...
-1
votes
2answers
598 views

The lie algebra of the orthogonal group of an arbitrary space time metric

Let X ad Y be two vectors in R4, and define the inner product of X and Y as: (X*Y) = gikXiYk (summation convention for repeated indicies) Then we consider the 4x4 matrix g whose components are gik. ...
5
votes
2answers
940 views

Jordan Form Over a Polynomial Ring

Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem ...
6
votes
3answers
2k views

Simultaneous diagonalization

I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ ...
12
votes
2answers
401 views

Matrices into path algebras

I was thinking about quivers recently, and the following idea came to me. Let ei,j denote the matrix unit in Mn for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, ...