NONCOMMUTATIVE GEOMETRY SEMINAR
Mathematical Institute of the Polish Academy of Sciences
Ul. Sniadeckich 8, room 408, Mondays, 10:15-12:00
1999/2001
2002/2003
2003/2004
2004/2005
3 October 2005
BRAIDED CYCLIC COHOMOLOGY
TOM HADFIELD
(Queen Mary, University of London, England)
10 October 2005
ON THE HOCHSCHILD HOMOLOGY OF QUANTUM SL(N)
It is shown that the standard quantized coordinate ring A:=k_q[SL(N)]
satisfies Van den Bergh's analogue of the Poincare duality for Hochschild
(co)homology. Here the dualizing bimodule is A_\sigma, ie the bimodule
that is A as a left A-module but with the right
multiplication twisted by the modular automorphism \sigma of the
Haar functional. This implies, in particular, H_{N^2-1}(A,A_\sigma)=k.
ULRICH KRAEHMER
(Instytut Matematyczny, Polska Akademia Nauk)
17 October 2005
VECTOR BUNDLES AND PROJECTIVE MODULES
In 1955, J.P. Serre showed that there is a one-to-one correspondence
between algebraic vector bundles over an affine variety and finitely
generated projective modules over its coordinate ring. Later, in 1961,
G. Swan showed analogous correspondence for topological vector
bundles over a compact Hausdorff space X and finitely generated
projective modules over the ring C(X) of continuous functions
on X (real, complex or quaternion - valued depending on the type
of vector bundles under consideration). This result can be extended
to the paracompact Hausdorff spaces provided we will restrict
ourselves to the bundles of finite type. In the talk I will present the
proof of the most general result due to L. Vaserstein
(Vector bundles and projective modules),
which holds
for an arbitrary topological space X. (In particular, it does not need
to be Hausdorff.) Of course, one has to give the correct definition
of the bundles of the finite type in the situation of arbitrary topological
spaces. The result can be formulated as an equivalence of categories
of finitely generated projective modules over C(X) and the category
of vector bundles of finite type over X.
In analogy with the
well-known case of compact Hausdorff spaces,
there is also a homotopical
classification of vector bundles of finite type over general spaces,
which will be sketched after L. Vaserstein.
PAWEL WITKOWSKI (Instytut Matematyki, Uniwersytet Warszawski)
24 October 2005
K-THEORETIC DUALITY FOR HIGHER RANK GRAPH ALGEBRAS
AND BOUNDARY CROSSED PRODUCTS
Classical Poincare duality says that there is a canonical
isomorphism between the cohomology and homology of a compact manifold
implemented by the cap-product with a certain fundamental class. Kasparov's
KK-theory allows to formulate an analogue of Poincare duality for
noncommutative C*-algebras. We establish such a duality result for a
class of higher rank graph algebras and discuss applications to duality for
groups acting on buildings.
(Based on
the joint work with Iulian Popescu.)
JOACHIM ZACHARIAS (The University of Nottingham, England)
7 November 2005
FUNDAMENTAL PROBLEMS OF NONCOMMUTATIVE TOPOLOGY
The Gelfand-Naimark duality between the category of commutative
unital C*-algebras and the category of compact Hausdorff spaces leads
to
various noncommutative eneralizations of topological constructions.
Also, it inspires interpretation
of some constructions with
noncommutative C*-algebras in a `dual topological' language. This approach
follows the following recipe. First,
take a property (construction, theorem etc.)
of compact Hausdorff spaces and dualize it to the category of commutative
unital C*-algebras. Then generalize it to noncommutative unital C*-algebras,
and finally dualize it to the opposite category of the category of
(noncommutative) unital C*-algebras. The latter category is viewed
as the category of `noncommutative compact Hausdorff spaces'. One
interpretes results obtained in the aforementioned way as a `noncommutative generalization' of
the initial topological property (construction, theorem etc.). A
pedagogical panorama of results and problems in this field will be
presented.
TOMASZ MASZCZYK (Instytut Matematyczny PAN, Instytut Matematyki UW)
14 November 2005
PROPER ACTIONS
PIOTR STACHURA
(Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)
21 November 2005
THE RELATIVE CHERN-GALOIS CHARACTER
The Chern-Galois theory is developed for corings or coalgebras
over non-commutative
rings. As the first step the notion of an entwined extension as
an extension of algebras
within a bijective entwining structure over a non-commutative ring is
introduced.
A strong connection for an entwined extension is defined and it is shown
to be closely related to the Galois property and to the equivariant
projectivity
of the extension. A generalisation of the Doi theorem on total integrals
in
the
framework of entwining structures over a non-commutative ring is
obtained, and
the bearing of strong connections on properties such as faithful flatness
or
relative injectivity is revealed. A family
of morphisms between
the K0-group of the category of finitely generated projective
comodules
of a coring
and even relative cyclic homology groups of the base algebra of
an entwined extension with a strong connection is constructed.
This is termed a relative Chern-Galois character. Explicit
examples include the computation of a Chern-Galois character of
depth 2 Frobenius split (or separable) extensions over
a separable algebra R.
Finitely generated and projective modules
are associated to an entwined extension with a
strong connection, the explicit form of idempotents is derived,
the corresponding (relative) Chern characters are computed, and
their connection with the relative Chern-Galois character is explained.
Joint work with Gabriella Bohm (Budapest).
TOMASZ BRZEZINSKI
(University of Wales, Swansea, Wales)
28 November 2005
VECTOR BUNDLES AS COTENSOR PRODUCTS
PIOTR M. HAJAC
(Instytut Matematyczny PAN / Katedra Metod Matematycznych Fizyki UW)
5 December 2005
QUANTUM HEISENBERG-LORENTZ GROUP
The aim of the seminar is to describe a general scheme of deforming
a classical group to its quantum version, called Rieffel deformation.
This deformation is based on twisting the Kac-Takesaki operator. In this
way, we obtain a new operator satisfing the pentagonal equation.
This leads to a quantum group. We apply this procedure to SL(2,C) and
describe the algebraic structure of the object constructed this way.
PAWEL L. KASPRZAK
(Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)