# 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.

3,985 questions
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### Uniqueness of the Lie brackets in the quotient space of a Lie algebra

Suppose I have a Lie algebra $\mathfrak g$ which has an ideal $\mathfrak a$. Then I consider the quotient set $\mathfrak g / \mathfrak a$ which is the set of all equivalence relations of $\mathfrak g$ ...
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### Quotient algebras of nilpotent Lie algebra are nilpotent

For the following proposition I found a proof in some notes that I don't understand. Below definition 1 defines the terminology I'm using, and proof attempt 1 gives my attempt at the proposition. ...
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### What is the definition of polynilpotent Lie algebras?

I am looking for definition of polynilpotent Lie algebras. Is there any equivalent concept for that?
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### Proving the Poisson differential is a coboundary map

Let $(M, \pi)$ be a Poisson manifold, that is, $M$ is a smooth real manifold and $\pi \in \mathfrak{X}^2(M)$ is a (possibly degenerate) skew-symmetric bivector field satisfying $[\pi, \pi] = 0$. Here ...
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### Simple Lie algebra representations and tensor powers of fundamental representations [duplicate]

Let $\frak{g}$ be a simple Lie algebra over $\mathbb{C}$. We will call a representation of $\frak{g}$ tautological if it is a fundamental representation of smallest dimension. For $V$ a tautological ...
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I'm trying to solve the following problem of the book "Grupos de Lie - Luiz A. B. San Martin": Question: Let $G$ be a connected Lie Group with Lie algebra $\mathfrak{g}$. Suppose that $X,Y \in \... 1answer 51 views ### What is the rigorous way for a Lie group (SU(n)) element to be “near” another element? Statement of the problem I'm working with a function$\lambda : SU(n)\times SU(n)\times SU(n) \rightarrow \mathbb{C}$. Given$U_1, U_2, U_3 \in SU(n)$, I'd like to know how to calculate$\lambda (\...
I want to prove the following: Given two representations of a connected matrix Lie group are equivalent if and only if the associated Lie algebra representations are equivalent. Definition: Let $G$ ...
$\newcommand{\ad}{\operatorname{ad}}$Let $R$ be an associative ring. Set $[x, y] = xy - yx$ and $\ad_x(-) = [x, -]: R \to R$. Is there a formula for $(ab)^n$ in general? I found one formula for the ...