# Questions tagged [lie-algebras]

For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

3,898 questions
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### For $G$ a real connected solvable Lie group, the commutator group $[G,G]$ is nilpotent.

I was stuck on showing the following problem: For $G$ a real connected solvable Lie group, the commutator group $[G,G]$ is nilpotent. There are a few ways I thought about this problem: Approach ...
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### Doubt in understading Proof of Matrix lie group and Lie algebra locally homemorphic

I was reading Brian C Hall Lie Group book In that I encountered following proof . I understand Whole proof. But have one doubt Why Auther take Orthogonal complement into consideration As I think ...
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### Every Borel contains a Cartan, and conjugacy theorems: A simple proof?

Conjugacy of Borel subalgebras $\newcommand{\ad}{\mathrm{ad}\,}$ Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. A Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$ is a ...
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### $\mathbb R$-points of semisimple real algebraic groups, connectivity, and Cartan involutions: some questions

I am reading about Cartan involutions on semisimple real Lie groups and have a point of confusion I am trying to reconcile with linear algebraic groups. Let $\mathbf G$ be a linear algebraic group ...
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### Deciding whether a representation is orthogonal or symplectic

I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement ...
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### Question about definition related to root system of semisimple Lie algebras

Let $L$ be a semisimple Lie algebra of finite dimension over a field of charcteristic 0 and algebraically closed, and $H$ a maximal toral subalgebra. Let $R$ be the set of roots of $L$ with respect ...
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### How to show a vector space is a real form of another?

I only know from the definition that, say $V_1$ is a real form of $V$, if $V=\mathbb{C}\otimes_\mathbb{R} V_1$, but what does this really mean? Is it true it is just saying that $V$ is spanned by ...
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### KIlling vectors from isometries and orbit spaces

I am currently (trying) to learn more about orbit spaces generated from an isometry group of a manifold. I cannot quite pinpoint what I (don't) understand, so I will try to lay out what I could gather:...
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### Relationship of SU(2) & SU(2) to SO(4)?

While almost all accessible references indicate/demonstrate that group SO(4) = SU(2)⊗SU(2), I've come across two references that state the relationship as SO(4) = SU(2)⊕SU(2). Is the latter equation ...