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.

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 KK-algebra AA with an element λK ZK 0(A)\lambda\in K\otimes_{\mathbf{Z}}K_0(A) a new algebra π λ(A)\pi^\lambda(A) and a canonical homomorphism Aπ λ(A)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)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 A_\infty-modules over Weyl algebra.

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

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.