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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.

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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
    
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
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@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
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