Skip to main content
SpringerLink
Log in

Indecomposables and Dimensions

  • Chapter
  • First Online:
Introduction to the Representation Theory of Algebras

  • 2563 Accesses

Abstract

Some of the deepest results in the theory of representations of algebras are presented within this chapter. All of them deal with the representation type, in other words, with the amount of non-isomorphic indecomposable modules with a fixed dimension vector. Several results will only be indicated without proof.

First the two Brauer-Thrall conjectures are discussed, but only the first is proven. Both of them deal with the various possibilities of having finitely or infinitely many non-isomorphic representations of a fixed dimension. Next, Gabriel’s Theorem is proved. It yields a complete characterization of those algebras which among the hereditary algebras are of finite representation type. The role of the associated quadratic form will play a crucial role.

Then Kac’s Theorem will be indicated and the distinction between the tame and wild representation type is discussed. It is shown why in the wild case a classification of all indecomposable modules must seem a hopeless endeavour. At the end a geometric approach is presented to consider the classification problem afresh.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Auslander, M.: Representation theory of Artin algebras II. Commun. Algebra 1, 269–310 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bautista, R.: On algebras of strongly unbounded representation type. Comment. Math. Helv. 60, 392–399 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bernstein, I.N., Gel’fand, I.M., Ponomarev, V.A.: Coxeter functors and Gabriel’s theorem. Russ. Math. Surv. 28, 17–32 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  4. Crawley-Boevey, W.W.: Matrix problems and Drozd’s theorem. Banach Center Publ. 26(1), 199–222 (1990)

    MathSciNet  Google Scholar 

  5. Drozd, J.: Tame and wild matrix problems. In: Representation Theory II. Lecture Notes in Mathematics, vol. 832, pp. 242–258. Springer, Berlin (1980)

    Google Scholar 

  6. Gabriel, P.: Unzerlegbare Darstellungen. I. (German) Manuscr. Math. 6, 71–103 (1972); correction, ibid. 6, 309 (1972)

    Google Scholar 

  7. Gabriel, P., Nazarova, L.A., Roiter, A.V., Sergeichuk, V.V., Vossieck, D.: Tame and wild subspace problems. Ukranian Math. J. 45, 335–372 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Humphreys, J.E.: Reflection Groups and Coxeter Groups, vol. 29. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  9. Kac, V.: Infinite root systems, representations of graphs and invariant theory I. Invent. Math. 56, 57–92 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kac, V.: Some remarks on representations of quivers and infinite root systems. Carleton Mathematical Lecture Notes, vol. 25, pp. 1301–1317 (1980)

    Google Scholar 

  11. King, A.D.: Moduli of representations of finite-dimensional algebras. Q. J. Math. Oxford Ser. 45(4), 515–530 (1994)

    Article  MATH  Google Scholar 

  12. Kraft, H.: Geometric methods in representation theory. In: Representations of Algebras. Lecture Notes in Mathematics, vol. 944, pp. 180–258. Springer, Berlin (1982)

    Google Scholar 

  13. Kraft, H. Riedtmann, C.: Geometry of representations of quivers. In: Webb, P. (ed.) Representations of Algebras. London Mathematical Society Lecture Note Series, vol. 116, pp. 109–145 (1986)

    MathSciNet  Google Scholar 

  14. Reineke, M.: Moduli of representations of quivers. In: Trends in Representation Theory of Algebras and Related Topics, EMS Publishing House 2008. Preprint (2008). arXiv:0802.2147v1

    Google Scholar 

  15. Ringel, C.M.: On algorithms for solving vector space problems. I. Report on the Brauer-Thrall conjectures: Roiter’s theorem and the theorem of Nazarova and Roiter. Representation theory, I (Proc. Workshop, Carleton University, Ottawa, ON, 1979). Springer LNM, vol. 831, pp. 104–136 (1980)

    MathSciNet  Google Scholar 

  16. Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, Berlin (1984)

    Google Scholar 

  17. Roiter, A.V.: Unboundedness of the dimension of the indecomposable representations of an algebra which has infinitely many indecomposable representations. Izv. Akad. Nauk SSSR. Ser. Mat. 32, 1275–1282 (1968)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Michael Barot

Authors
  1. Michael Barot
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Barot, M. (2015). Indecomposables and Dimensions. In: Introduction to the Representation Theory of Algebras. Springer, Cham. https://doi.org/10.1007/978-3-319-11475-0_9

Download citation

Publish with us

Policies and ethics

Search

Navigation