# Tagged Questions

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### regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
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### Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
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### Codimension of $\ker$ $\alpha$

Can someone explain why the codimension of $\ker$ $\alpha$ is $1$ in $H$, with complement $Fh_\alpha$? Is this because $h_\alpha$ when $\alpha$ is simple is part of the dual basis to ...
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### Dual spaces: Roots and Cartan subalgebra

Can someone show that the roots and the Cartan subalgebra are dual vector spaces? I don't see how simple roots acting on non-corresponding indices of a Cartan basis produce 0 and a simple root ...
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### Associative Lie algebra without Jacobi identity

1) Is there a name for associative Lie algebra that does not require Jacobi identity to hold? 2) Can such algebra exist, and if it does exist, can this algebra contain infinitely many elements? 3) ...
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### two Roots questions

Just two questions on roots... 1) Can the length of roots only be defined relatively? And does length only come about because of the dot product and cartan integers? 2) This might be a weird ...
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### Closure of a Fundamental Weyl Chamber

Can someone explain what a "closure" of a Fundamental Weyl Chamber means? I assume it is related to an algebraic closure, but I don't see how. In addition, how does the Weyl group act on it and why ...
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### killing form and the dot product

When going from talking about roots as functionals to talking about roots as vectors in a Euclidian space (root system), does the killing form become the dot product? Are the killing form and dot ...
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### Borel subalgebras contain solvable radical

Let $L$ be a Lie algebra and let $B$ be a Borel subalgebra (a maximal solvable subalgebra) of $L$. I want to understand why $\operatorname{Rad} L \subseteq B$. In his proof, Humphreys ...
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### Problem 9.7 - Lie Algebras - Humphreys

Let $\alpha,\beta\in\Phi$ span a subspace $E'$ of $E$. Prove that $E'\cap\Phi$ is a root system in $E'$. Prove similarly that $\Phi\cap(\mathbb{Z}\alpha+\mathbb{Z}\beta)$ is a root system in E' (must ...
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### Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

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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### Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
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### Extend to a homomorphism.

My question is regarding a step in "p-Automorphisms of Finite p-Groups" by Evgenii I. Khukhro (p. 117 line 7) and would like some response to my argumentation/understanding of it. p-Automorphisms of ...
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### The decomposition of the exterior of the symmetric square over Lie algebra sl(3)

I am studying the representation theory of finite dimensional modules over the simple Lie algebra $\operatorname{sl}(3)$. I know some basics facts about the decomposition of some construction of ...
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### Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
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### Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
### Invertible derivation of $[L, \operatorname{Rad}(L)]$
Suppose $L$ is a finite-dimensional Lie algebra over the field of characteristic $0$. Let $\operatorname{Rad}(L)$ denote the radical of $L$. My question is: Does there always exist an invertible ...