MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Are there any definitions of quivers for algebras which are not basic or unital? I am reading the book Elements of the representation theory of associative algebras: volume one. The ordinary quivers and AR quivers are defined only for unital basis algebras. If I have an algebra which is not basic or not unital, could we compute some kinds of quivers for this algebra?

I search on the Internent and find that there are some quivers called Ext-quivers. Are there any other quivers? What are the relations between these quivers? Thank you very much.

share|cite|improve this question
2  
For unital non-basic algebras there is a unique up to isomorphism unital basic algebra which is Morita equivalent to it and one uses that algebras quivers. I don't have a good answer for the nonunital case unless you want to add a unit. – Benjamin Steinberg Apr 13 '13 at 14:44
1  
My guess is that in the non-basic case one could start with the quiver for a Morita-equivalent basic algebra (as described above by B. Steinberg), then interpret each vertex $v_i$ as representing a $k_i \times k_i$ matrix algebra, each edge as representing a rectangular $k_i \times k_j$ matrix of elements of the radical, and so on. I don't know of a reference for this construction, and I would be very interested to know if there is one. – Kevin Walker Apr 13 '13 at 16:39
1  
@Kevin: I haven't looked at it very closely, but I think they are doing something like that in section 3 of arXiv:1303.7049 – Dag Oskar Madsen Apr 13 '13 at 21:31
    
@Dag: Yes, that does look like what I had in mind -- thanks! – Kevin Walker Apr 14 '13 at 4:07

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.