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An idempotent for a Jordanian quantum complex sphere

100%
Zieliński B.
Banach Center Publications
|
2003
|
tom 61
|
nr 1
345-348
EN
A new Jordanian quantum complex 4-sphere together with an instanton-type idempotent is obtained as a suspension of the Jordanian quantum group $SL_h(2)$.
2
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Cocycle condition for multi-pullbacks of algebras

63%
Hajac P., Zieliński B.
Banach Center Publications
|
2012
|
tom 98
|
nr 1
239-243
EN
Take finitely many topological spaces and for each pair of these spaces choose a pair of corresponding closed subspaces that are identified by a homeomorphism. We note that this gluing procedure does not guarantee that the building pieces, or the gluings of some pieces, are embedded in the space obtained by putting together all given ingredients. Dually, we show that a certain sufficient condition, called the cocycle condition, is also necessary to guarantee sheaf-like properties of surjective multi-pullbacks of algebras with distributive lattices of ideals.
3
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Finite closed coverings of compact quantum spaces

51%
Hajac P., Kaygun A., Zieliński B.
Banach Center Publications
|
2012
|
tom 98
|
nr 1
215-237
EN
We consider the poset of all non-empty finite subsets of the set of natural numbers, use the poset structure to topologise it with the Alexandrov topology, and call the thus obtained topological space $ℙ^∞$ the universal partition space. Then we show that it 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 partition 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. In particular, the Gelfand transform allows us to view finite closed coverings of compact Hausdorff spaces as flabby sheaves of commutative unital C*-algebras over $ℙ^∞$.
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