I try to prove the following theorem which given without proof here On Prime Ideals of Lie Algebras

Theorem:Let $L$ and $L^{prime}$ be Lie algebras and let $f: L rightarrow L^{prime}$ be a surjective homomorphism. Then an ideal $P$ of $L$ containing $mathrm{Ker} f$ is prime if and only if $f(P)$ is prime in $L^{prime}$.

My Proof:Let $P$ be a prime ideal of $L$, let $H$ and $K$ are two ideals of $L^{prime}$. Suppose that $[H,K] subseteq f(P)$, then $f^{-1}([H,K]) subseteq f^{-1}f(P)$, hence $Big [f^{-1}(H),f^{-1}(K)Big] subseteq P$.

But $P$ is prime, then $f^{-1}(H) subseteq P$ or $f^{-1}(K) subseteq P$. Since $f$ is surjective homomorphism, thus

$H subseteq f(P) textit{ or } K subseteq f(P)$. Therefore $f(P)$ is prime.

Is this proof true?

Is this step $f^{-1}([H,K])= Big [f^{-1}(H),f^{-1}(K)Big]$ true?

Is this step $f^{-1}f(P)=P$ true?

Is this theorem is a trivial conclusion of Isomorphism Theorems of Lie algebra and we do not need this proof?

Mathematics Asked by Hamada Al on November 12, 2021

1 AnswersYou have correctly identified the weak points of your argument:

(1) Is this step $f^{-1}([H,K])= Big [f^{-1}(H),f^{-1}(K)Big]$ true?

No, in general this is not true. If the Lie bracket on $L$ was zero, then the right hand side would be $0$, but the left hand side would contain ker$(f)$.

However for your purpose all you need is: $$f^{-1}([H,K])supseteq Big [f^{-1}(H),f^{-1}(K)Big],$$ which is true. To see this just apply $f$ to an element of the right hand side, and clearly you will land in $[H,K]$.

(2) Is this step $f^{-1}f(P)=P$ true?

Yes, but only because of one specific bit of information you were given, which you should mention to justify the statement. Namely ker$(f)subseteq P$. Thus if $xin f^{-1}f(P)$ then $f(x)=f(p)$ for some $p in P$, and $$x=(x-p)+p,$$ with $pin P$ and $x-pin$ker$(f)subseteq P$.

Answered by tkf on November 12, 2021

2 Asked on February 22, 2021 by taylor-rendon

1 Asked on February 22, 2021

3 Asked on February 22, 2021 by believer

2 Asked on February 22, 2021 by inquirer

0 Asked on February 22, 2021 by user237522

algebraic geometry commutative algebra maximal and prime ideals polynomials ring theory

2 Asked on February 22, 2021

algebraic geometry algebraic number theory commutative algebra integral extensions modules

2 Asked on February 22, 2021 by wyatt

1 Asked on February 22, 2021

2 Asked on February 22, 2021 by user330477

1 Asked on February 22, 2021

5 Asked on February 21, 2021 by arshiya

1 Asked on February 21, 2021 by michael-maier

1 Asked on February 21, 2021 by user160

1 Asked on February 21, 2021 by mashed-potato

0 Asked on February 21, 2021 by melisa-karimi

2 Asked on February 21, 2021 by annie-marie-heart

group theory lie algebras lie groups representation theory semi riemannian geometry

0 Asked on February 21, 2021 by tuong-nguyen-minh

1 Asked on February 20, 2021 by stanislas-castellana

1 Asked on February 20, 2021 by canine360

Get help from others!

Recent Questions

- Constraints on $x^2 + b x + c = 0$ such that at least one root has a positive real part
- Picking the set of coset representatives of a stabilizer under a group action
- Write condition if rectangle is inside another rectangle
- How to segment a group of symmetric points
- Prove $D(A):= {win H^{2}(0, 2pi): w(0) = w(2pi), w^{‘}(0) = w^{‘}(2pi) }$ is dense in $L^{2}(0,2pi).$

Recent Answers

- NeitherNor on How to segment a group of symmetric points
- Ethan Bolker on A is greater than B by 25% then by what percentage B is less than A?
- zkutch on A is greater than B by 25% then by what percentage B is less than A?
- lab bhattacharjee on Constraints on $x^2 + b x + c = 0$ such that at least one root has a positive real part
- Dominique on A is greater than B by 25% then by what percentage B is less than A?

© 2021 InsideDarkWeb.com. All rights reserved.