# Two generators

1. May 3, 2006

### semisimple

Hi,
I have been asked to prove that an algebra with only two generators can not be simple. I have thought about it and have tried to show that one of the two structure constants go to zero in the Lie algebra condition that [xi,xj]=sumoverk Cijkxk , but I can't find any way to show that (using the Bianci Identity, for example), and perhaps I am thinking about it the wrong way. I have also tried to think about it logically - perhaps it contains the identity for some reason and any algebra containing the identity is not simple.
I would appreciate any hints people might give me.

2. May 4, 2006

### matt grime

You can't mean just 'algebra', can you? The quaternionic fields are algebras and have 2 generators. They are skew fields, hence simple (no ideals), indeed they are central simple algebras, aren't they?

Perhaps you mean 'Lie Algebra'? Indeed you must mean something other than 'algebra' by your last assertion that any algebra with an identity is not simple (the full matrix algebra of nxn matrices over some field is simple and has an identity).

Simple lie algebras are copies of sl_2 glued together. That's one starting point, I don't know how far it'll get you.

Last edited: May 4, 2006
3. May 4, 2006

### semisimple

Hi Matt,
From what you say it must be Lie Algebras that I have to prove it for. The question just said "prove that an algebra...", but the title of the questions in general says "Lie Algebras". I was unsure at first, but decided he must mean Lie Algebra for that question. Sorry I didn't specify. I will think about sl2 idea.