Skip to main content
Log in

Infinite dimensional representations of finite dimensional algebras and amenability

  • Published:
Mathematische Annalen Aims and scope Submit manuscript


We present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra R over a countable field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on R-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part. We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930’s. As an application in algorithm theory, we introduce and study the notion of parameter testing of modules over finite dimensional algebras, that is analogous to the testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Arzhantseva, G., Paunescu, L.: Linear sofic groups and algebras. (preprint)

  2. Bartholdi, L.: On amenability of group algebras. Israel J. Math. 168, 153–165 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barot, M.: Introduction to the Representation Theory of Algebras. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  4. Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23) (2001). (electronic)

  5. Brown, C., Chuang, J., Lazarev, A.: Derived localization of algebras and modules. (preprint)

  6. Butler, M.C.R., Ringel, C.M.: Auslander-Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15(1–2), 145–179 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Capraro, V., Capraro, M.: Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture, Lecture Notes in Mathematics, vol. 2136. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  8. Cohn, P.M.: Free Rings and Their Relations. London Mathematical Society Monographs, vol. 19. Academic Press, Inc., London (1985)

    Google Scholar 

  9. Cohn, P.M.: Rank functions on projective modules over rings. Contemporary Mathematics, vol. 131, Part 2, pp. 271–277. American Mathematical Society, Providence, RI (1992)

  10. Cohn, P.M.: The universal skew field of fractions of a tensor product of free rings. Colloq. Math. 72(1), 1–8 (1997)

    MathSciNet  MATH  Google Scholar 

  11. Crawley-Boevey, W.: Infinite dimensional modules in the representation theory of finite dimensional algebras. In: Algebras and Modules, I (Trondheim, 1996), CMS Conference Proceedings, vol. 23, pp. 29-54. American Mathematical Society, Providence, RI (1998)

  12. Elek, G.: The amenability and non-amenability of skew fields. Proc. Am. Math. Soc. 134(3), 637–644 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Elek, G.: Parameter testing in bounded degree graphs of subexponential growth. Random Struct. Algorithms 37(2), 248–270 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Elek, G.: Finite graphs and amenability. J. Funct. Anal. 263(9), 2593–2614 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Elek, G.: Convergence and limits of linear representations of finite groups. J. Algebra 450, 588–615 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goldreich, O., Ron, D.: Property testing in bounded degree graphs. Algorithmica 32(2), 302–343 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goodearl, K.R.: von Neumann Regular Rings. Robert E. Krieger Publishing Co. Inc., Malabar, FL (1991)

    MATH  Google Scholar 

  18. Gromov, M.: Topological invariants of dynamical systems and spaces of holomorphic maps. I. Math. Phys. Anal. Geom. 2(4), 323–415 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Halperin, I.: von Neumann’s manuscript on inductive limits of regular rings. Can. J. Math. 20, 477–483 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hassidim, A., Kelner, J.A., Nguyen, H.N., Onak, K.: Local graph partitions for approximation and testing. In: 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 22–31 (2009)

  21. Lovász, L.: Large networks and graph limits. American Mathematical Society Colloquium Publications, vol. 60. American Mathematical Society, Providence, RI (2012)

  22. Newman, I., Sohler, C.: Every property of hyperfinite graphs is testable. SIAM J. Comput. 42(3), 1095–1112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Prest, M.: Purity, Spectra and Localisation. Encyclopedia of Mathematics and its Applications, vol. 121. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  24. Ringel, C.M.: Infinite-dimensional representations of finite-dimensional hereditary algebras. Symposia Mathematica, Vol. XXIII (Conference Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), pp. 321–412. Academic Press, London-New York (1979)

  25. Schofield, A.H.: Representations of Rings over Skew Fields, Lecture Notes, vol. 92. London Mathematics Society (1985)

  26. Schramm, O.: Hyperfinite graph limits. Electron. Res. Announc. Math. Sci. 15, 17–23 (2008)

    MathSciNet  MATH  Google Scholar 

  27. von Neumann, J.: Examples of continuous geometries. Proc. Nat. Acad. Sci. USA 22, 101–108 (1936)

    Article  MATH  Google Scholar 

  28. Ziegler, M.: Model theory of modules. Ann. Pure Appl. Logic 26(2), 149–213 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references


The author is grateful for the hospitality of the Bernoulli Center at the École Polytechnique Fédérale Lausanne , where some of this work was carried out.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Gábor Elek.

Additional information

Research partly sponsored by MTA Renyi “Lendulet” Groups and Graphs Research Group.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elek, G. Infinite dimensional representations of finite dimensional algebras and amenability. Math. Ann. 369, 397–439 (2017).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification