Introduced by Gelfand and Ponomarev in study of representations of quivers
I. M. Gelfand, V. A. Ponomarev, Model algebras and representations of graphs:, Funkc. Anal. i Priložen. 13 (1979) 1–12. Engl.transl. Func. Anal. Appl. 13 (1979) 157–166.
When a path algebra of the quiver is replaced by a more general finite dimensional hereditary -algebra , the construction of the algebra structure on a representative system of indecomposable left (right) -modules is proposed in
D. Baer, W. Geigle, H. Lenzing, The preprojective algebra of a tame hereditary Artin algebra, Commun. Algebra 15 (1987), 425–457 doi
The deformed version is described in
W. Crawley-Boevey, Martin P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (3): 605–635 (1998) doiMR1620538
A construction generalizing deformed preprojective algebra of quivers and assigning to a -algebra with an element a new algebra and a canonical homomorphism is described in
W. Crawley-Boevey, Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities, Comment. Math. Helv. 74 (1999) 548-574 doi
Some examples of this construction are the algebra of differential operators on a smooth curve in characteristic zero and the cotangent bundle of . Conze’s original construction is for an embedding of a Weyl algebra. Modules over deformed preprojective algebras are in some case closely related to -modules over Weyl algebra.
Yuri Berest, Calogero-Moser spaces over algebraic curves, Sel. math., New ser. 14, 373–396 (2009) doiarXiv:0809.4521
Yuri Berest, Oleg Chalykh, Farkhod Eshmatov, Recollement of deformed preprojective algebras and the Calogero-Moser correspondence, Mosc. Math. J. 8:1 (2008) 21–37