2016/12/05 by Nguyen, Van C., Todorov, Gordana, Zhu, Shijie
#16G20 #16G70 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1612.01585
Let Q be a tree-type quiver, k Q its path algebra, and λ a nonzero element in the field k. We construct irreducible morphisms in the Auslander-Reiten quiver of the transjective component of the bounded derived category of k Q that satisfy what we call the λ-relations. When λ=1, the relations are known as mesh relations. When λ=-1, they are known as commutativity relations. Using this technique together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree-type quiver, several descriptions of its preprojective algebra are equivalent.