# 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.

406 questions
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### On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
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### Which Lie groups have Lie algebras admitting an Ad-invariant inner product?

I am trying to answer the following question: Which Lie groups have a Lie algebra admitting an $\text{Ad}$-invariant inner product? First of all, all compact Lie groups satisfy this condition ...
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### Showing the Lie Algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$ are not isomorphic.

I am working through the exercises in "Lie Groups, Lie Algebras, and Representations" - Hall and can't complete exercise 11 of chapter 3. My aim was to demonstrate that there does not exist a vector ...
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### Under what conditions is the exponential map on a Lie algebra injective?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\exp :\mathfrak{g}\rightarrow G$ be the exponential map. In his blog, Terence Tao notes that if a Lie group is not simply-connected, ...
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### Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra. My motivation is that I (eventually) want to understand the theory underpinning papers ...
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### Classsifying 1- and 2- dimensional Algebras, up to Isomorphism

I am trying to find all 1- or 2- dimensional Lie Algebras "a" up to isomorphism. This is what I have so far: If a is 1-dimensional, then every vector (and therefore every tangent vector field) is of ...
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### Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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### Is every element of a complex semisimple Lie algebra a commutator?

Let $L$ be a (finite-dimensional) complex semisimple Lie algebra. Then we know that $L = [L,L]$. Is it true that every element of $L$ must be a commutator? Since a complex semisimple Lie algebra is ...
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### Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)\times SU(2)$, or their Lie algebras

I have seen sources claim that $SO^+(1,3) \cong SU(2) \times SU(2)$, but have seen others claim that only their Lie algebras are isomorphic. Is it true that $SO^+(1,3) \cong SU(2) \times SU(2)$? If ...
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I'm looking to solve some matrix equations. One of the equations involves a commutator, so my question is as follows: let $A$ be a skew-self-adjoint, traceless matrix, does the equation $[X,Y] = A$ ...
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### Solvable Lie algebra with codimension 1 ideal

There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal". The proof I am thinking of is the following. ...
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### Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
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### Expression of unitary group , the discrete subgroups and invariants

Let $$G=U(3),$$ be the unitary group. Here we consider $G$ in terms of the fundamental representation of U(3). Namely, all of $g \in G$ can be written as a rank-3 (3 by 3) matrices. What is the ...
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### Subgroups and invariants in a unitary group U(3)

This is related to the post, but an enriched version of the problem. Now we require the richer form of $P_1,P_2,P_3,P_4,P_5,P_6$. Let $$G=U(3),$$ be the unitary group. Here we consider $G$ in terms ...
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### Does the abstract Jordan decomposition agree with the usual Jordan decomposition in a semisimple Lie subalgebra of endomorphisms?

Is it true that for every element $x$ of a semisimple Lie subalgebra of endomorphisms $L\subseteq \text{End}(V)$, where $V$ is a finite dimensional vector space over $\mathbb{C}$, the abstract Jordan ...
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### Non-Abelian subgroups and invariants in a unitary group 2

Related to a previous question, let us redefine their $\{P_1, P_2, P_3, -P_1, - P_2, - P_3 \}$ to something somehow different. Let $$G=U(3),$$ be the unitary group. Here we consider $G$ in terms of ...
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### $gl(2,\mathbb C) \cong sl(2,\mathbb C) \oplus \mathbb C$

Hi I just start learning Lie algebra and there is one hw question I don't really understand how to do, hope somebody give me some hints. $L_1,L_2$ are Lie algebras. $L=\{(x_1,x_2):x_i \in L_i\}$. Lie ...
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### Semisimple Lie algebras are perfect.

Can anyone explain why a semi-simple finite dimensional Lie algebra $\mathfrak{g}$ has to be perfect ? The natural way to prove something like that would be to look to the algebra generated by the ...
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### Computing information about a Lie algebra from cartan matrix

Let's consider the Cartan matrix : $$\begin{pmatrix} 2 & -2 \\ -1 & 2 \end{pmatrix}$$ I am asked to find the number of roots and then to compute character of the adjoint representation. ...
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### Show that the characters of the representations $\phi_{n}$ of $SU(2)$ constitute a complete orthogonal set.

The question is given below: And the other questions mentioned are (I know the solutions of all of them): Sorry for the bad formulation of the my question at the first time I have ...
### Strictly Upper Triangular $n\times n$ Matrices form a Nilpotent Lie Algebra over $\mathbb{C}$ for $n\geq 2$
I have been attempting this problem for a while, it is an assignment problem so I don't want somebody to just post the answer, I'm just looking for hints. Let $\mathfrak{u}(n,\mathbb{C})$ be the Lie ...