Results

**1 - 1**of**1**### Preprojective algebras and Calabi-Yau duality

"... The properties of the preprojective algebra are very different whether the associ-ated quiver is of Dynkin type or not. However in both cases, one can construct from it a triangulated category of Calabi-Yau dimension 2. In this note we explain the generalizations of this fact in the context of highe ..."

Abstract
- Add to MetaCart

(Show Context)
The properties of the preprojective algebra are very different whether the associ-ated quiver is of Dynkin type or not. However in both cases, one can construct from it a triangulated category of Calabi-Yau dimension 2. In this note we explain the generalizations of this fact in the context of higher preprojective algebra, and we give some homological properties that characterize preprojective algebras. 1. Classical case Let k = k be an algebraically closed field. Let Q be a finite quiver. The double quiver Q of Q is defined from Q by adding for each arrow a ∈ Q1 an arrow a in the opposite direction. The preprojective algebra of Q is defined by ΠQ: = kQ/〈 a∈Q1 [a, a]〉. This notion has been defined by Gelfand and Ponomarev in [9]. Example 1.1. Let Q be the following quiver 1 x zz. Then we have kQ ∼ = k[x]. The preprojective algebra of Q is presented by the quiver 1 x zz x with the relation xx − xx = 0. That is ΠQ ∼ = k[x, x]. Example 1.2. Let Q be the quiver 1 a / / 2. Then the preprojective algebra of Q is presented by the quiver 1 a