Generalisations of Preprojective algebras

In this thesis, we investigate two ways of generalising the preprojective algebra. First, we introduce the multiplicative preprojective algebra, Λ(Q), which is a multiplicative analogue of the deformed preprojective algebra, introduced by Crawley-Boevey and Holland. The special case q = 1 is the undeformed multiplicative preprojective algebra, which is an analogue of the ordinary (undeformed) preprojective algebra. We adapt the middle convolution operation of Dettweiler and Reiter to construct reflection functors, which are used to determine the possible dimension vectors of simple modules for Λ(Q). We show that Λ(Q) is finite dimensional if Q is Dynkin, and that e1Λ (Q)e1 is a commutative integral domain of Krull dimension 2 if Q is extended Dynkin with 1 an extending vertex. The proofs of these results depend on applying the reduction algorithm as described by Bergman, which is recalled in the appendix. We conjecture that the undeformed multiplicative preprojective algebra is a ‘preprojective algebra’ in the sense of Gelfand and Ponomarev, in that as a KQ-module, it is isomorphic to the direct sum of the indecomposable preprojective KQ-modules. Second, we extend the notion of a preprojective algebra of a quiver to the notion of a preprojective algebra for a quiver with relations. Our results show that for any Nakayama algebra A, there exists an algebra P (A) such that P (A) is isomorphic to the direct sum of all indecomposable A-modules.