Let $L$ be a restricted Lie algebra with the restricted enveloping algebra $u(L)$ over a field $F$. Let $ω(L)$ denote the augmentation ideal of $u(L)$ which is the kernel of the augmentation map $\varepsilon : u(L) \mapsto F$ induced by $x\mapsto 0$. My question is as follows: Why the augmentation ideal is generated by $L$? May you please give an example?
Asked
Active
Viewed 144 times
1
-
Do you see that $L$ is contained in the kernel? – Tobias Kildetoft May 05 '16 at 07:14
-
why is $L$ in kernel? Indeed , I want to know more about what augmentation map does? – Nil May 05 '16 at 07:17
-
You wrote yourself that this is the map induced by $x\mapsto 0$. I assume you know the precise formulation of this part? – Tobias Kildetoft May 05 '16 at 07:23
-
yes, I wrote that according to the formal definition, the thing make me confused is that other elements of $u(L)$ which does not map to $0$? – Nil May 05 '16 at 07:28
-
No, the formal definition is not just $x\mapsto 0$, but includes a condition on what those $x$ can be. – Tobias Kildetoft May 05 '16 at 07:32
-
May you please clarify it more? – Nil May 05 '16 at 07:34
-
1You really need to look up the precise definition. Without that there is not much I can do to help. – Tobias Kildetoft May 05 '16 at 07:35
-
I would be grateful if you could introduce a book including this notion. – Nil May 05 '16 at 08:00
-
But where did you see it? – Tobias Kildetoft May 05 '16 at 08:05
-
I saw it in an article, it was a very brief statement. http://www.ams.org/journals/proc/1995-123-10/S0002-9939-1995-1264829-0/S0002-9939-1995-1264829-0.pdf – Nil May 05 '16 at 08:44
-
Sure, papers tend to assume the reader is familiar with the background material. Not sure what a good reference is for restricted Lie algebras though. Were there no references in the paper? – Tobias Kildetoft May 05 '16 at 08:46
-
I looked for proper reference within this article, I could not find . – Nil May 05 '16 at 08:55
-
Which paper is it? – Tobias Kildetoft May 05 '16 at 10:15
-
Restricted Lie algebras and their envelopes – Nil May 05 '16 at 10:20
-
The very last reference on the list is to a book which seems like it should be a good place to start. – Tobias Kildetoft May 05 '16 at 10:25
-
I have already started reading some notations related to group rings, I suppose I need to find out the way we correspond universal enveloping algebras to group ring? – Nil May 05 '16 at 11:29