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.*

Deformed version is described in

- W. Crawley-Boevey, Martin P. Holland,
*Noncommutative deformations of Kleinian singularities*, Duke Math. J. 92 (3): 605–635 (1998) doi MR1620538

A construction generalizing deformed preprojective algebra of quivers and assigning to a $K$-algebra $A$ with an element $\lambda\in K\otimes_{\mathbf{Z}}K_0(A)$ a new algebra $\pi^\lambda(A)$ and a canonical homomorphism $A\to \pi^\lambda(A)$ 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 $Spec(A)$. Conze’s original construction is for an embedding of a Weyl algebra. Modules over deformed preprojective algebras are in some case closely related to $A_\infty$-modules over Weyl algebra.

- Yuri Berest,
*Calogero-Moser spaces over algebraic curves*, Sel. math., New ser. 14, 373–396 (2009) doi arXiv:0809.4521 - Yuri Berest, Oleg Chalykh, Farkhod Eshmatov,
*Recollement of deformed preprojective algebras and the Calogero-Moser correspondence*, Mosc. Math. J.**8**:1 (2008) 21–37arXiv:0706.3006 doi MR2422265 mathnet.ru/mmj2

A point of view on preprojective algebras is a part of a picture in

- William Crawley-Boevey, Pavel Etingof, Victor Ginzburg,
*Noncommutative geometry and quiver algebras*, Adv. Math.**209**:1 (2007) 274-336 doi

Other contributions

- Tristan Bozec, Damien Calaque, Sarah Scherotzke,
*Calabi-Yau structures for multiplicative preprojective algebras*, arXiv:2102.12336 - Travis Schedler,
*Zeroth Hochschild homology of preprojective algebras over the integers*, Adv. Math.**299**(2016) 451–542 doi - Daniel Kaplan, Travis Schedler,
*Multiplicative preprojective algebras are 2-Calabi-Yau*, arXiv:1905.12025 - Christof Geiß, Bernard Leclerc, Jan Schroer,
*Rigid modules over preprojective algebras*, Invent. Math.**165**(3):589–632 (2006) doi - C. M. Ringel,
*The preprojective algebra of a quiver. Algebras and modules, II*, (Geiranger 1996) 467–480, CMS Conf. Proc.**24**

category: algebra

Last revised on September 17, 2023 at 12:49:05. See the history of this page for a list of all contributions to it.