# Tagged Questions

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### A criteria for a subalgebra of M(n,C) being M(n,C) [migrated]

Suppose $S$ is a subalgebra of the matrix algebra $M_n(\mathbb{C})$. If for any vector $v$ and $w$ in $\mathbb{C}$, there always exists a matrix $A$ in $S$, depending on $v$ and $w$ of course, which ...
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### Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...