Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Tensor products of Leavitt path algebras


Authors: Pere Ara and Guillermo Cortiñas
Journal: Proc. Amer. Math. Soc. 141 (2013), 2629-2639
MSC (2010): Primary 16E40, 16S99; Secondary 19D50
DOI: https://doi.org/10.1090/S0002-9939-2013-11561-3
Published electronically: April 25, 2013
MathSciNet review: 3056553
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We compute the Hochschild homology of Leavitt path algebras over a field $ k$. As an application, we show that $ L_2$ and $ L_2\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular, they are not isomorphic. Similarly, $ L_\infty $ and $ L_\infty \otimes L_\infty $ are distinguished by their Hochschild homologies, and so they are not Morita equivalent either. By contrast, we show that $ K$-theory cannot distinguish these algebras; we have $ K_*(L_2)=K_*(L_2\otimes L_2)=0$ and $ K_*(L_\infty )=K_*(L_\infty \otimes L_\infty )=K_*(k)$.


References [Enhancements On Off] (What's this?)

  • 1. G. Abrams, G. Aranda Pino. The Leavitt path algebra of a graph. J. Algebra 293 (2005) 319-334. MR 2172342 (2007b:46085)
  • 2. G. Abrams, A. Louly, E. Pardo, C. Smith. Flow invariants in the classification of Leavitt path algebras. J. Algebra 333 (2011) 202-231. MR 2785945
  • 3. P. Ara, M. Brustenga. Module theory over Leavitt path algebras and $ K$-theory. J. Pure Appl. Algebra 214 (2010) 1131-1151. MR 2586992 (2011b:16109)
  • 4. P. Ara, M. Brustenga. The regular algebra of a quiver. J. Algebra 309 (2007) 207-235. MR 2301238 (2008a:16019)
  • 5. P. Ara, M. Brustenga, G. Cortiñas. K-theory of Leavitt path algebras. Münster Journal of Mathematics 2 (2009) 5-34. MR 2545605 (2011d:46145)
  • 6. P. Ara, M.A. González-Barroso, K.R. Goodearl, E. Pardo. Fractional skew monoid rings.
    J. Algebra 278 (2004) 104-126. MR 2068068 (2005f:16042)
  • 7. P. Ara, M. A. Moreno, E. Pardo, Nonstable K-theory for graph algebras. Algebr. Represent. Theory 10 (2007) 157-178. MR 2310414 (2008b:46094)
  • 8. J. Bell, G. Bergman. Private communication, 2011.
  • 9. G. M. Bergman and Warren Dicks, Universal derivations and universal ring constructions. Pacific J. Math. 79 (1978) 293-337. MR 531320 (81b:16024)
  • 10. H. Cartan, S. Eilenberg, Homological Algebra. Princeton University Press, Princeton, N. J., 1956. MR 0077480 (17:1040e)
  • 11. G. Cortiñas, E. Ellis. Isomorphism conjectures with proper coefficients. |arXiv:1108.5196v3|.
  • 12. S. Eilenberg, A. Rosenberg, D. Zelinsky, On the dimension of modules and algebras, VIII. Dimension of tensor products. Nagoya Math. J. 12 (1957) 71-93. MR 0098774 (20:5229)
  • 13. S. M. Gersten. K-theory of free rings. Comm. Algebra 1 (1974) 39-64. MR 0396671 (53:533)
  • 14. E. Kirchberg, The classification of purely infinite C*-algebras using Kasparov theory. Preprint.
  • 15. E. Kirchberg, N.C. Phillips, Embedding of exact $ C^*$-algebras in the Cuntz algebra $ \mathcal O_2$.
    J. Reine Angew. Math. 525 (2000) 17-53. MR 1780426 (2001d:46086a)
  • 16. M. Lorenz. On the homology of graded algebras. Comm. Algebra 20 (1992) 489-507. MR 1146311 (93b:19003)
  • 17. W. G. Leavitt. The module type of a ring. Trans. Amer. Math. Soc. 103 (1962) 113-130. MR 0132764 (24:A2600)
  • 18. J. L. Loday, Cyclic homology. Grund. Math. Wiss., 301. Springer-Verlag, Berlin, Heidelberg, 1998. MR 1600246 (98h:16014)
  • 19. N. C. Phillips, A classification theorem for nuclear purely infinite simple C*-algebras. Doc. Math. 5 (2000) 49-114. MR 1745197 (2001d:46086b)
  • 20. I. Raeburn. Graph algebras. CBMS Regional Conference Series in Mathematics, 103. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2005. MR 2135030 (2005k:46141)
  • 21. M. Rørdam, A short proof of Elliott's theorem. C. R. Math. Rep. Acad. Sci. Canada 16 (1994) 31-36. MR 1276341 (95d:46064)
  • 22. M. Rørdam, Classification of nuclear, simple C*-algebras. Classification of nuclear C*-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., 126, 1-145, Springer, Berlin, 2002. MR 1878882 (2003i:46060)
  • 23. C. Weibel,
    An introduction to homological algebra,
    Cambridge Univ. Press, 1994. MR 1269324 (95f:18001)
  • 24. M. Wodzicki,
    Excision in cyclic homology and in rational algebraic $ K$-theory.
    Ann. of Math. 129 (1989) 591-639. MR 997314 (91h:19008)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16E40, 16S99, 19D50

Retrieve articles in all journals with MSC (2010): 16E40, 16S99, 19D50


Additional Information

Pere Ara
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Email: para@mat.uab.cat

Guillermo Cortiñas
Affiliation: Departamento de Matemática and Instituto Santaló, Ciudad Universitaria Pab 1, 1428 Buenos Aires, Argentina
Email: gcorti@dm.uba.ar

DOI: https://doi.org/10.1090/S0002-9939-2013-11561-3
Received by editor(s): August 1, 2011
Received by editor(s) in revised form: November 9, 2011
Published electronically: April 25, 2013
Additional Notes: The first author was partially supported by DGI MICIIN-FEDER MTM2008-06201-C02-01 and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
The second author was supported by CONICET and partially supported by grants PIP 112-200801-00900, UBACyTs X051 and 20020100100386, and MTM2007-64074.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society