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**1 - 4**of**4**### MODULI STACKS OF SERRE STABLE REPRESENTATIONS IN TILTING THEORY

"... Abstract. We introduce a new moduli stack, called the Serre stable moduli stack, which corresponds to studying families of point objects in an abelian category with a Serre functor. This allows us in particular, to re-interpret the classical derived equivalence between most concealed-canonical algeb ..."

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Abstract. We introduce a new moduli stack, called the Serre stable moduli stack, which corresponds to studying families of point objects in an abelian category with a Serre functor. This allows us in particular, to re-interpret the classical derived equivalence between most concealed-canonical algebras and weighted projective lines by showing they are induced by the universal sheaf on the Serre stable moduli stack. We explain why the method works by showing that the Serre stable moduli stack is the tautological moduli problem that allows one to recover certain nice stacks such as weighted projective lines from their moduli of sheaves. As a result, this new stack should be of interest in both representation theory and algebraic geometry. Throughout, we work over an algebraically closed base field k of characteristic zero. 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 ..."

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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