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1
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The garden of quantum spheres

100%
Dąbrowski L.
Banach Center Publications
|
2003
|
tom 61
|
nr 1
37-48
EN
A list of known quantum spheres of dimension one, two and three is presented.
2
Content available remote

Dirac operator on the standard Podleś quantum sphere

63%
Dąbrowski L., Sitarz A.
Banach Center Publications
|
2003
|
tom 61
|
nr 1
49-58
EN
Using principles of quantum symmetries we derive the algebraic part of the real spectral triple data for the standard Podleś quantum sphere: equivariant representation, chiral grading γ, reality structure J and the Dirac operator D, which has bounded commutators with the elements of the algebra and satisfies the first order condition.
3
Content available remote

Noncommutative Borsuk-Ulam-type conjectures

51%
Baum P., Dąbrowski L., Hajac P.
Banach Center Publications
|
2015
|
tom 106
|
nr 1
9-18
EN
Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if $δ: A → A ⨂_{min} H$ is a free coaction of the C*-algebra H of a non-trivial compact quantum group on a unital C*-algebra A, then there is no H-equivariant *-homomorphism from A to the equivariant join C*-algebra $A ⊛_δ H$. For A being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on ℤ/2ℤ, we recover the celebrated Borsuk-Ulam Theorem. The second conjecture states that there is no H-equivariant *-homomorphism from H to the equivariant join C*-algebra $A ⊛_δ H$. We show how to prove the conjecture in the special case $A = C(SU_q(2)) = H$, which is tantamount to showing the non-trivializability of Pflaum's quantum instanton fibration built from $SU_q(2)$.
4
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Index pairings for pullbacks of C*-algebras

45%
Dąbrowski L., Hadfield T., Hajac P., Matthes R., Wagner E.
Banach Center Publications
|
2012
|
tom 98
|
nr 1
67-84
EN
In this overview, we study how to reduce the index pairing for a fibre-product C*-algebra to the index pairing for the C*-algebra over which the fibre product is taken. As an example we analyze the case of suspensions and apply it to noncommutative instanton bundles of arbitrary charges over the suspension of quantum deformations of the 3-sphere.
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