89 research outputs found
Noncommutative Instantons on the 4-Sphere from Quantum Groups
We describe an approach to the noncommutative instantons on the 4-sphere
based on quantum group theory. We quantize the Hopf bundle S^7 --> S^4 making
use of the concept of quantum coisotropic subgroups. The analysis of the
semiclassical Poisson--Lie structure of U(4) shows that the diagonal SU(2) must
be conjugated to be properly quantized. The quantum coisotropic subgroup we
obtain is the standard SU_q(2); it determines a new deformation of the 4-sphere
Sigma^4_q as the algebra of coinvariants in S_q^7. We show that the quantum
vector bundle associated to the fundamental corepresentation of SU_q(2) is
finitely generated and projective and we compute the explicit projector. We
give the unitary representations of Sigma^4_q, we define two 0-summable
Fredholm modules and we compute the Chern-Connes pairing between the projector
and their characters. It comes out that even the zero class in cyclic homology
is non trivial.Comment: 16 pages, LaTeX; revised versio
Representations and cohomology for Frobenius-Lusztig kernels
Let be the quantum group (Lusztig form) associated to the simple
Lie algebra , with parameter specialized to an -th
root of unity in a field of characteristic . In this paper we study
certain finite-dimensional normal Hopf subalgebras of ,
called Frobenius-Lusztig kernels, which generalize the Frobenius kernels
of an algebraic group . When , the algebras studied here reduce to the
small quantum group introduced by Lusztig. We classify the irreducible
-modules and discuss their characters. We then study the
cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent
and Borel subalgebras corresponding to unipotent and Borel subgroups of . We
prove that the cohomology ring for the first Frobenius-Lusztig kernel is
finitely-generated when \g has type or , and that the cohomology rings
for the nilpotent and Borel subalgebras are finitely-generated in general.Comment: 26 pages. Incorrect references fixe
Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures
We develop the basic constructions of homological algebra in the
(appropriately defined) unbounded derived categories of modules over algebras
over coalgebras over noncommutative rings (which we call semialgebras over
corings). We define double-sided derived functors SemiTor and SemiExt of the
functors of semitensor product and semihomomorphisms, and construct an
equivalence between the exotic derived categories of semimodules and
semicontramodules.
Certain (co)flatness and/or (co)projectivity conditions have to be imposed on
the coring and semialgebra to make the module categories abelian (and the
cotensor product associative). Besides, for a number of technical reasons we
mostly have to assume that the basic ring has a finite homological dimension
(no such assumptions about the coring and semialgebra are made).
In the final sections we construct model category structures on the
categories of complexes of semi(contra)modules, and develop relative
nonhomogeneous Koszul duality theory for filtered semialgebras and
quasi-differential corings.
Our motivating examples come from the semi-infinite cohomology theory.
Comparison with the semi-infinite (co)homology of Tate Lie algebras and graded
associative algebras is established in appendices, and the semi-infinite
homology of a locally compact topological group relative to an open profinite
subgroup is defined. An application to the correspondence between complexes of
representations of an infinite-dimensional Lie algebra on the complementary
central charge levels ( and for the Virasoro) is worked out.Comment: Dedicated to the memory of my father. LaTeX 2e, 310 pages. With
appendices coauthored by S.Arkhipov and D.Rumynin. v.12: changes in the
Introduction, additions to Section 0 and Appendix D, small improvements in
Appendix C and elsewhere, subtitle added -- this is intended as the final
arXiv version; v.13: abstract updated, LaTeX file unchanged (the publisher's
version is more complete
Banach Lie-Poisson spaces and reduction
The category of Banach Lie-Poisson spaces is introduced and studied. It is
shown that the category of W*-algebras can be considered as one of its
subcategories. Examples and applications of Banach Lie-Poisson spaces to
quantization and integration of Hamiltonian systems are given. The relationship
between classical and quantum reduction is discussed.Comment: 58 pages, to apear in Comm.Math.Phy
Nilpotence and descent in equivariant stable homotopy theory
Let be a finite group and let be a family of subgroups of
. We introduce a class of -equivariant spectra that we call
-nilpotent. This definition fits into the general theory of
torsion, complete, and nilpotent objects in a symmetric monoidal stable
-category, with which we begin. We then develop some of the basic
properties of -nilpotent -spectra, which are explored further
in the sequel to this paper.
In the rest of the paper, we prove several general structure theorems for
-categories of module spectra over objects such as equivariant real and
complex -theory and Borel-equivariant . Using these structure theorems
and a technique with the flag variety dating back to Quillen, we then show that
large classes of equivariant cohomology theories for which a type of
complex-orientability holds are nilpotent for the family of abelian subgroups.
In particular, we prove that equivariant real and complex -theory, as well
as the Borel-equivariant versions of complex-oriented theories, have this
property.Comment: 63 pages. Revised version, to appear in Advances in Mathematic
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