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Leavitt Path Algebras of Hypergraphs

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Abstract

We define Leavitt path algebras of hypergraphs generalizing simultaneously Leavitt path algebras of separated graphs and Leavitt path algebras of vertex-weighted graphs (i.e. weighted graphs that have the property that any two edges emitted by the same vertex have the same weight). We investigate the Leavitt path algebras of hypergraphs in terms of linear bases, Gelfand–Kirillov dimension, ring-theoretic properties (e.g. simplicity, von Neumann regularity and Noetherianess), K-theory and graded K-theory. By doing so we obtain new results on the Gelfand–Kirillov dimension and graded K-theory of Leavitt path algebras of separated graphs and on the graded K-theory of Leavitt path algebras of vertex-weighted graphs.

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References

  1. Abrams, G., Ara, P., Siles, Molina, M.: Leavitt path algebras, Lecture Notes in Mathematics 2191, Springer, Berlin (2017)

  2. Abrams, G., Pino, G.A.: The Leavitt path algebra of a graph. J. Algebra 293(2), 319–334 (2005)

  3. Ara, P., Hazrat, R., Li, H., Sims, A.: Graded Steinberg algebras and their representations. Algebra Number Theory 12(1), (2018)

  4. Ara, P., Goodearl, K.R.: Leavitt path algebras of separated graphs. J. Reine angew. Math. 669, 165–224 (2012)

  5. Ara, P., Moreno, M.A., Pardo, E.: Nonstable \(K\)-theory for graph algebras. Algebr. Represent. Theory 10(2), 157–178 (2007)

  6. Bergman, G.M.: Coproducts and some universal ring constructions. Trans. Am. Math. Soc. 200, 33–88 (1974)

  7. Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29(2), 178–218 (1978)

  8. Cohn, P.M.: On the free product of associative rings. III. J. Algebra 8, 376–383 (1968)

  9. Cuntz, J.: Simple \(C^*\)-algebras generated by isometries. Comm. Math. Phys. 57(2), 173–185 (1977)

  10. Hazrat, R.: The graded structure of Leavitt path algebras. Israel J. Math. 195(2), 833–895 (2013)

  11. Hazrat, R., Preusser, R.: Applications of normal forms for weighted Leavitt path algebras: simple rings and domains. Algebr. Represent. Theor. 20, 1061–1083 (2017)

  12. Leavitt, W.G.: Modules over rings of words. Proc. Am. Math. Soc. 7, 188–193 (1956)

  13. Leavitt, W.G.: Modules without invariant basis number. Proc. Am. Math. Soc. 8, 322–328 (1957)

  14. Leavitt, W.G.: The module type of a ring. Trans. Am. Math. Soc. 103, 113–130 (1962)

  15. Leavitt, W.G.: The module type of homomorphic images. Duke Math. J. 32, 305–311 (1965)

  16. Moreno-Fernandez, J.M., Siles-Molina, M.: Graph algebras and the Gelfand-Kirillov dimension. J. Algebra Appl. 17(5), 1850095 (2018)

  17. Phillips, N.C.: A classification theorem for nuclear purely infinite simple \(C^*\)-algebras. Doc. Math. 5, 49–114 (2000)

  18. Preusser, R.: The V-monoid of a weighted Leavitt path algebra. Israel J. Math. (2018, accepted)

  19. Preusser, R.: The Gelfand-Kirillov dimension of a weighted Leavitt path algebra. J. Algebra Appl. (2019, accepted)

  20. Raeburn, I.: Graph algebras. In: CBMS Regional Conference Series in Mathematics, 103. American Mathematical Society, Providence, RI (2005)

  21. Small, L.W., Warfield Jr., R.B.: Prime affine algebras of Gelfand-Kirillov dimension one. J. Algebra 91, 386–389 (1984)

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Correspondence to Raimund Preusser.

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Preusser, R. Leavitt Path Algebras of Hypergraphs. Bull Braz Math Soc, New Series 51, 185–221 (2020). https://doi.org/10.1007/s00574-019-00150-3

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Keywords

  • Leavitt path algebras
  • Hypergraphs
  • K-theory

Mathematics Subject Classification

  • 16S10
  • 16W10
  • 16W50
  • 16D70