P. Baum and J. Brodzki :
Equivariant \( KK \) -theory and noncommutative index theory 2004 .
Part VI of e-book “Lecture notes on noncommutative geometry and quantum groups” (European Mathematical Society, ed. Piotr M. Hajac).
misc
People
BibTeX
@misc {key64605687,
AUTHOR = {Baum, Paul and Brodzki, J.},
TITLE = {Equivariant \$KK\$-theory and noncommutative
index theory},
HOWPUBLISHED = {Part VI of e-book ``Lecture notes on
noncommutative geometry and quantum
groups'' (European Mathematical Society,
ed. Piotr M. Hajac)},
YEAR = {2004},
PAGES = {613--706},
URL = {http://www.mimuw.edu.pl/~pwit/toknotes/toknotes.pdf},
}
P. Baum and R. Meyer :
The Baum–Connes conjecture, localization of categories and quantum groups 2004 .
Part VIII of e-book “Lecture notes on noncommutative geometry and quantum groups” (European Mathematical Society, ed. Piotr M. Hajac).
misc
People
BibTeX
@misc {key56409780,
AUTHOR = {Baum, Paul and Meyer, R.},
TITLE = {The {B}aum--{C}onnes conjecture, localization
of categories and quantum groups},
HOWPUBLISHED = {Part VIII of e-book ``Lecture notes
on noncommutative geometry and quantum
groups'' (European Mathematical Society,
ed. Piotr M. Hajac)},
YEAR = {2004},
PAGES = {867--952},
URL = {http://www.mimuw.edu.pl/~pwit/toknotes/toknotes.pdf},
}
P. Baum and H. Moscovici :
Foliations, \( C^* \) -algebras and index theory 2004 .
Part II of e-book “Lecture notes on noncommutative geometry and quantum groups” (European Mathematical Society, ed. Piotr M. Hajac).
misc
People
BibTeX
@misc {key69857717,
AUTHOR = {Baum, Paul and Moscovici, H.},
TITLE = {Foliations, \$C^*\$-algebras and index
theory},
HOWPUBLISHED = {Part II of e-book ``Lecture notes on
noncommutative geometry and quantum
groups'' (European Mathematical Society,
ed. Piotr M. Hajac)},
YEAR = {2004},
PAGES = {135--245},
URL = {http://www.mimuw.edu.pl/~pwit/toknotes/toknotes.pdf},
}
P. F. Baum, P. M. Hajac, R. Matthes, and W. Szymański :
“The \( K \) -theory of Heegaard-type quantum 3-spheres \( K \) -Theory35 : 1–2
(2005 ),
pp. 159–186 .
Dedicated to the memory of Olaf Richter.
An erratum to this was published in K-Theory 37 :1–2 (2006)
MR
2240219 Zbl
1111.46051 article
Abstract
People
BibTeX
@article {key2240219m,
AUTHOR = {Baum, Paul F. and Hajac, Piotr M. and
Matthes, Rainer and Szyma\'nski, Wojciech},
TITLE = {The \$K\$-theory of {H}eegaard-type quantum
3-spheres},
JOURNAL = {\$K\$-Theory},
FJOURNAL = {\$K\$-Theory. An Interdisciplinary Journal
for the Development, Application, and
Influence of \$K\$-Theory in the Mathematical
Sciences},
VOLUME = {35},
NUMBER = {1--2},
YEAR = {2005},
PAGES = {159--186},
DOI = {10.1007/s10977-005-1550-y},
NOTE = {Dedicated to the memory of Olaf Richter.
An erratum to this was published in
\textit{K-Theory} \textbf{37}:1--2 (2006).
MR:2240219. Zbl:1111.46051.},
ISSN = {0920-3036},
}
P. F. Baum, P. M. Hajac, R. Matthes, and W. Szymański :
“Erratum: ‘The \( K \) -theory of Heegaard-type quantum 3-spheres’ \( K \) -Theory37 : 1–2
(2006 ),
pp. 211 .
Erratum to an article published in K-Theory 35 :1–2 (2005)
MR
2274673 Zbl
1210.46054 article
People
BibTeX
@article {key2274673m,
AUTHOR = {Baum, Paul F. and Hajac, Piotr M. and
Matthes, Rainer and Szyma\'nski, Wojciech},
TITLE = {Erratum: ``{T}he \$K\$-theory of {H}eegaard-type
quantum 3-spheres''},
JOURNAL = {\$K\$-Theory},
FJOURNAL = {\$K\$-Theory. An Interdisciplinary Journal
for the Development, Application, and
Influence of \$K\$-Theory in the Mathematical
Sciences},
VOLUME = {37},
NUMBER = {1--2},
YEAR = {2006},
PAGES = {211},
DOI = {10.1007/s10977-006-0026-z},
NOTE = {Erratum to an article published in \textit{K-Theory}
\textbf{35}:1--2 (2005). MR:2274673.
Zbl:1210.46054.},
ISSN = {0920-3036},
}
P. Baum, P. M. Hajac, R. Matthes, and W. Szymanski :
Non-commutative geometry approach to principal and associated bundles .
Preprint ,
2007 .
ArXiv
math/0701033v2 techreport
Abstract
People
BibTeX
We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) \( C^* \) -algebras. We introduce the concept of piecewise triviality to adapt the standard notion of local triviality to fibre products of \( C^* \) -algebras. In the context of principal actions, we study in detail an example of a non-proper free action with continuous translation map, and examples of compact principal bundles which are piecewise trivial but not locally trivial, and neither piecewise trivial nor locally trivial, respectively. We show that the module of continuous sections of a vector bundle associated to a compact principal bundle is a cotensor product of the algebra of functions defined on the total space (that are continuous along the base and polynomial along the fibres) with the vector space of the representation. On the algebraic side, we review the formalism of connections for the universal differential algebras. In the differential geometry framework, we consider smooth connections on principal bundles as equivariant splittings of the cotangent bundle, as 1-form-valued derivations of the algebra of smooth functions on the structure group, and as axiomatically given covariant differentiations of functions defined on the total space. Finally, we use the Dirac monopole connection to compute the pairing of the line bundles associated to the Hopf fibration with the cyclic cocycle of integration over \( S^2 \) .
@techreport {keymath/0701033v2a,
AUTHOR = {Baum, Paul and Hajac, Piotr M. and Matthes,
Rainer and Szymanski, Wojciech},
TITLE = {Non-commutative geometry approach to
principal and associated bundles},
TYPE = {preprint},
YEAR = {2007},
NOTE = {ArXiv:math/0701033v2.},
}
P. F. Baum and P. M. Hajac :
“Local proof of algebraic characterization of free actions Special issue on noncommutative geometry and quantum groups in honor of Marc A. Rieffel ,
published as SIGMA
10 .
Issue edited by G. Elliott, P. M. Hajac, H. Li, and J. Rosenberg .
2014 .
paper no. 060.
MR
3226990 Zbl
1295.22010 ArXiv
1402.3024 incollection
Abstract
People
BibTeX
Let \( G \) be a compact Hausdorff topological group acting on a compact Hausdorff topological space \( X \) . Within the \( C^* \) -algebra \( C(X) \) of all continuous complex-valued functions on \( X \) , there is the Peter–Weyl algebra \( \mathcal{P}_G(X) \) which is the (purely algebraic) direct sum of the isotypical components for the action of \( G \) on \( C(X) \) . We prove that the action of \( G \) on \( X \) is free if and only if the canonical map
\[ \mathcal{P}_G(X)\otimes^{}_{C(X/G)}\mathcal{P}_G(X)\to\mathcal{P}_G(X)\otimes\mathcal{O}(G) \]
is bijective. Here both tensor products are purely algebraic, and \( \mathcal{O}(G) \) denotes the Hopf algebra of “polynomial” functions on \( G \) .
@article {key3226990m,
AUTHOR = {Baum, Paul F. and Hajac, Piotr M.},
TITLE = {Local proof of algebraic characterization
of free actions},
JOURNAL = {SIGMA},
FJOURNAL = {Symmetry, Integrability and Geometry.
Methods and Applications},
VOLUME = {10},
YEAR = {2014},
DOI = {10.3842/SIGMA.2014.060},
NOTE = {\textit{Special issue on noncommutative
geometry and quantum groups in honor
of {M}arc {A}. {R}ieffel}. Issue edited
by G. Elliott, P. M. Hajac,
H. Li, and J. Rosenberg.
paper no. 060. ArXiv:1402.3024. MR:3226990.
Zbl:1295.22010.},
ISSN = {1815-0659},
}
P. F. Baum, L. Dąbrowski, and P. M. Hajac :
“Noncommutative Borsuk–Ulam-type conjectures 9–18
in
From Poisson brackets to universal quantum symmetries
(Warsaw, 18–22 August 2014 ).
Edited by N. Ciccoli and A. Sitarz .
Banach Center Publications 106 .
Instytut Matematyczny PAN (Warsaw ),
2015 .
MR
3469159 Zbl
1343.46064 incollection
Abstract
People
BibTeX
Within the framework of free actions of compact quantum groups on unital \( C^* \) -algebras, we propose two conjectures. The first one states that, if
\[ \delta:A\to A\otimes_{\min}H \]
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 \( A\circledast_\delta H \) . For \( A \) being the \( C^* \) -algebra of continuous functions on a sphere with the antipodal coaction of the \( C^* \) -algebra of functions on \( \mathbb{Z}/2\mathbb{Z} \) , 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 \( A\circledast_\delta H \) . 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 \( SU_q(2) \) .
@incollection {key3469159m,
AUTHOR = {Baum, Paul F. and D\polhk abrowski,
Ludwik and Hajac, Piotr M.},
TITLE = {Noncommutative {B}orsuk--{U}lam-type
conjectures},
BOOKTITLE = {From {P}oisson brackets to universal
quantum symmetries},
EDITOR = {Ciccoli, Nicola and Sitarz, Andrzej},
SERIES = {Banach Center Publications},
NUMBER = {106},
PUBLISHER = {Instytut Matematyczny PAN},
ADDRESS = {Warsaw},
YEAR = {2015},
PAGES = {9--18},
DOI = {10.4064/bc106-0-1},
NOTE = {(Warsaw, 18--22 August 2014). MR:3469159.
Zbl:1343.46064.},
ISSN = {0137-6934},
ISBN = {9788386806294},
}
P. F. Baum, K. De Commer, and P. M. Hajac :
“Free actions of compact quantum groups on unital \( C^* \) -algebras Doc. Math.
22
(2017 ),
pp. 825–849 .
MR
3665403 Zbl
06810396 ArXiv
1304.2812v1 article
Abstract
People
BibTeX
Let \( F \) be a field, \( \Gamma \) a finite group, and \( \operatorname{Map}(\Gamma,F) \) the Hopf algebra of all set-theoretic maps \( \Gamma\to F \) . If \( E \) is a finite field extension of \( F \) and \( \Gamma \) is its Galois group, the extension is Galois if and only if the canonical map
\[ E\otimes_F E\to E\otimes_F \operatorname{Map}(\Gamma,F) \]
resulting from viewing \( E \) as a \( \operatorname{Map}(\Gamma,F) \) -comodule is an isomorphism. Similarly, a finite covering space is regular if and only if the analogous canonical map is an isomorphism. In this paper, we extend this point of view to actions of compact quantum groups on unital \( C^* \) -algebras. We prove that such an action is free if and only if the canonical map (obtained using the underlying Hopf algebra of the compact quantum group) is an isomorphism. In particular, we are able to express the freeness of a compact Hausdorff topological group action on a compact Hausdorff topological space in algebraic terms. As an application, we show that a field of free actions on unital \( C^* \) -algebras yields a global free action.
@article {key3665403m,
AUTHOR = {Baum, Paul F. and De Commer, Kenny and
Hajac, Piotr M.},
TITLE = {Free actions of compact quantum groups
on unital \$C^*\$-algebras},
JOURNAL = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
VOLUME = {22},
YEAR = {2017},
PAGES = {825--849},
URL = {https://www.math.uni-bielefeld.de/documenta/vol-22/23.pdf},
NOTE = {ArXiv:1304.2812v1. MR:3665403. Zbl:06810396.},
ISSN = {1431-0635},
}
