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Locally finite Leavitt path algebras

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Abstract

A group-graded K-algebra A = ⊕ g∈G A g is called locally finite in case each graded component A g is finite dimensional over K. We characterize the graphs E for which the Leavitt path algebra L K (E) is locally finite in the standard ℤ-grading. For a locally finite ℤ-graded algebra A we show that, if every nonzero graded ideal has finite codimension in A, then every nonzero ideal has finite codimension in A; that is, ℤ-graded just infinite implies just infinite. We use this result to characterize the finite graphs E for which the Leavitt path algebra L K (E) is locally finite just infinite. We then give an explicit description of the graphs and algebras which arise in this way. In particular, we show that the locally finite Leavitt path algebras are precisely the noetherian Leavitt path algebras.

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References

  1. G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, Journal of Algebra 293 (2005), 319–334.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Abrams and G. Aranda Pino, Purely infinite simple Leavitt path algebras, Journal of Pure and Applied Algebra 207 (2006), 553–563. DOI: 10.1016/j.jpaa.2005.10.010.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Abrams, G. Aranda Pino and M. Siles Molina, Finite dimensional Leavitt path algebras, Journal of Pure and Applied Algebra 209 (2007), 753–762.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Ara, M. A. Moreno and E. Pardo, Nonstable K-Theory for graph algebras, Algebra Represent. Theory 10 (2007), 157–178.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Aranda Pino, E. Pardo and M. Siles Molina, Exchange Leavitt path algebras and stable rank, Journal of Algebra 305 (2006), 912–936. DOI: 10.1016/j.jalgebra.2005.12.009.

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Aranda Pino, F. Perera and M. Siles Molina, eds., Graph algebras: bridging the gap between analysis and algebra, University of Málaga Press, Malaga, Spain, 2007. ISBN: 978-84-9747-177-0

    Google Scholar 

  7. J. Farina and C. Pendergrass-Rice, A few properties of just infinite algebras, Communications in Algebra 35 (2007), 1703–1707.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Farkas and L. Small, Algebras which are nearly finite dimensional and their identities, Israel Journal of Mathematics 127 (2002), 245–251.

    Article  MATH  MathSciNet  Google Scholar 

  9. T. Y. Lam, Lectures on Modules and Rings, Graduate texts in Mathematics, Vol 189, Springer-Verlag, New York, 1999.

    MATH  Google Scholar 

  10. W. G. Leavitt, The module type of a ring, Trans. A.M.S. 42 (1962), 113–130.

    Article  MathSciNet  Google Scholar 

  11. I. Raeburn, Graph algebras. CBMS Regional Conference Series in Mathematics, vol 103, American Mathematical Society, Providence, 2005.

    MATH  Google Scholar 

  12. Z. Reichstein, D. Rogalski and J. J. Zhang, Projectively simple rings, Advances in Mathematics 203 (2006), 365–407. Available at arxiv.math.RA/0401098.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Colorado Springs, CO, 80933, USA

    G. Abrams

  2. Departamento de Álgebra, Universidad Complutense de Madrid, 28040, Madrid, Spain

    G. Aranda Pino

  3. Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29071, Málaga, Spain

    M. Siles Molina

Authors
  1. G. Abrams
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  2. G. Aranda Pino
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  3. M. Siles Molina
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Correspondence to G. Abrams.

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Abrams, G., Aranda Pino, G. & Siles Molina, M. Locally finite Leavitt path algebras. Isr. J. Math. 165, 329–348 (2008). https://doi.org/10.1007/s11856-008-1014-1

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  • DOI: https://doi.org/10.1007/s11856-008-1014-1

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