Hi. In Assem's book et al "Elements of Representation Theory I" it is an exercise to classify all 3 dimensional basic, connected K-algebras where K is an algebraically closed field. Unless I'm wrong I get that the answer is that there are only three such algebras, my question is: what about classification of higher dimensional basic connected K-algebras, is this known?
1 Answer
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The hereditary ones are all path algebras of quivers. More generally, these algebras can be classified by using quivers and relations.
There are some very nice lecture notes by Bill Crawley-Boevey, at http://www1.maths.leeds.ac.uk/~pmtwc/ and by Bernhard Keller at, http://people.math.jussieu.fr/~keller/#Enseignement (and probably many others...).