nLab preprojective algebra

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 $k$-algebra $A$, the construction of the algebra structure on a representative system of indecomposable left (right) $A$-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) 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) 21–37

  arXiv: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, J. Noncommut. Geom. 17(3) (2023) 783–810 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