About Universal enveloping algebra

[HDB1]

Please, I have a question about this:​

The Universal enveloping algebra of a finite dimensional Lie algebra is Noetherian.

How we can prove it? Please..

 

[HDB1]

Dear @fresh_42 , I am so sorry for bothering you, please, if you could hlep, i would appreciate it..

 

[fresh_42]

Please, I have a question about this:​

The Universal enveloping algebra of a finite dimensional Lie algebra is Noetherian.

How we can prove it? Please..

A module that is also a vector space is Noetherian if and only if it is finite-dimensional. The universal enveloping algebra is both, a module, and a vector space. We must therefore show that the universal enveloping algebra of a finite-dimensional Lie algebra is finite-dimensional, too. This is the statement of the Poincaré-Birkhoff-Witt theorem, proven by Humphreys (GTM 9) in chapters 17.3 and 17.4., Corollary 17.3.C.

 

[HDB1]

Thank you so much, @fresh_42 , please, why Universal enveloping algebra is module? PBW theorem gives a basis of Universal enveloping algebra, but please, why it is finite dimensional? please,

I thougt in general: lie lagebra is finite dimensioal, and its universal enveloping is infinite dimensional.

Thanks in advance,

 

[fresh_42]

Thank you so much, @fresh_42 , please, why Universal enveloping algebra is module? PBW theorem gives a basis of Universal enveloping algebra, but please, why it is finite dimensional? please,

I thougt in general: lie lagebra is finite dimensioal, and its universal enveloping is infinite dimensional.

Thanks in advance,

It is a $\mathbb{K}$-vector space and as such a $\mathbb{K}$-module. We say vector space and finite-dimensional in case the scalars are from a field, and we say module and finitely generated in case the scalars are from a ring, e.g. the integers.

The question is: How do you define Noetherian? It is usually defined for rings and modules. E.g. a module is Noetherian if it is finitely generated. But finitely generated modules over a ring that is a field like in our case, are automatically finite-dimensional vector spaces. And PBW makes sure that the universal enveloping algebra of a finite-dimensional Lie algebra is again finite-dimensional.