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Atabey Kaygun

  1. Jacobi-Zariski Exact Sequence for Hochschild Homology and Cyclic (Co)Homology.

    Thu, 03/24/2011 - 16:01 — SomNambul
    Authors: Atabey Kaygun
    Subjects: K-Theory and Homology
    Abstract

    We prove that for an inclusion of unital associative but not necessarily
    commutative algebras $A\subseteq B$ we have long exact sequences in Hochschild
    homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in
    Andr\'e-Quillen homology, provided that the quotient $B$-module $A/B$ is flat.
    We also prove that for an arbitrary r-flat morphism $f:B\to A$ with an H-unital
    kernel, one can express the Wodzicki excision sequence and the corresponding
    Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a
    single long exact sequence.

  2. Quantum projective space from Toeplitz cubes.

    Thu, 08/05/2010 - 15:01 — SomNambul
    Authors: Piotr M. Hajac, Atabey Kaygun, Bartosz Zielinski
    Subjects: Quantum Algebra
    Abstract

    From N-tensor powers of the Toeplitz algebra, we construct a multipullback
    C*-algebra that is a noncommutative deformation of the complex projective space
    CP(N). Using Birkhoff's Representation Theorem, we prove that the lattice of
    kernels of the canonical projections on components of the multipullback
    C*-algebra is free. This shows that our deformation preserves the freeness of
    the lattice of subsets generated by the affine covering of the complex
    projective space.

  3. Finite closed coverings of compact quantum spaces.

    Thu, 10/29/2009 - 16:01 — SomNambul
    Authors: Piotr M. Hajac, Atabey Kaygun, Bartosz Zielinski
    Subjects: Quantum Algebra
    Abstract

    We show that a projective space P^\infty(Z/2) endowed with the Alexandrov
    topology is a classifying space for finite closed coverings of compact quantum
    spaces in the sense that any such a covering is functorially equivalent to a
    sheaf over this projective space. In technical terms, we prove that the
    category of finitely supported flabby sheaves of algebras is equivalent to the
    category of algebras with a finite set of ideals that intersect to zero and
    generate a distributive lattice.

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