JournalsjncgVol. 15, No. 1pp. 113–146

Algebraic bivariant KK-theory and Leavitt path algebras.

  • Guillermo Cortiñas

    Universidad de Buenos Aires, Argentina
  • Diego Montero

    Universidad de Buenos Aires, Argentina
Algebraic bivariant $K$-theory and Leavitt path algebras. cover
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Abstract

We investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras L(E)L(E) and L(F)L(F) of graphs EE and FF over a commutative ground ring \ell. We approach this by studying the structure of such algebras under bivariant algebraic KK-theory kkkk, which is the universal homology theory with the properties above. We show that under very mild assumptions on \ell, for a graph EE with finitely many vertices and reduced incidence matrix AEA_E, the structure of L(E)L(E) in kkkk depends only on the groups Coker(IAE)(I-A_E) and Coker(IAEt)(I-A_E^t). We also prove that for Leavitt path algebras, kkkk has several properties similar to those that Kasparov's bivariant KK-theory has for CC^*-graph algebras, including analogues of the Universal coefficient and Künneth theorems of Rosenberg and Schochet.

Cite this article

Guillermo Cortiñas, Diego Montero, Algebraic bivariant KK-theory and Leavitt path algebras.. J. Noncommut. Geom. 15 (2021), no. 1, pp. 113–146

DOI 10.4171/JNCG/397