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1.
We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres of Connes and Landi and of Connes and Dubois Violette, by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives, by a quotient procedure, a meaningful calculus on the sphere. In this calculus, the external algebra has the same dimension as the classical one. We develop the Haar functional on spheres and use it to define an integral of forms. In the twisted limit (differently from the general multiparametric case), the Haar functional is a trace and we thus obtain a cycle on the algebra. Moreover, we explicitly construct the *-Hodge operator on the space of forms on the plane and then by quotient on the sphere. We apply our results to even spheres and compute the Chern–Connes pairing between the character of this cycle, i.e. a cyclic 2n-cocycle, and the instanton projector defined in math.QA/0107070. 相似文献
2.
De Rham Cohomology and Hodge Decomposition For Quantum Groups 总被引:1,自引:0,他引:1
Let be one of the N2-dimensionalbicovariant first order differential calculi for the quantumgroups GLq(N), SLq(N), SOq(N), or Spq(N), where q is a transcendentalcomplex number and z is a regular parameter. It is shown thatthe de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologiesof its left-coinvariant, its right-coinvariant and its (two-sided)coinvariant subcomplexes. In the cases GLq(N) and SLq(N) thecohomology ring is isomorphic to the coinvariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in thesecases. The main technical tool is the spectral decompositionof the quantum Laplace-Beltrami operator. 2000 MathematicalSubject Classification: 46L87, 58A12, 81R50. 相似文献
3.
István Heckenberger 《Compositio Mathematica》2000,123(3):329-354
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace–Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite-dimensional. Using Jucys–Murphy elements of the Hecke algebra, the eigenvalues of the Laplace–Beltrami operator for the Hopf algebra
(SL
q
(N)) are computed. 相似文献
4.
We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups k(M)
k
G associated to finite group factorizations X = GM and a field k. The irreducible calculi are associated to certain conjugacy classes in X and representations of isotropy groups. We find the full exterior algebras and show that they are inner by a bi-invariant 1-form which is a generator in the noncommutative de Rham cohomology H
1. The special cases where one subgroup is normal are analysed. As an application, we study the noncommutative cohomology on the quantum codouble D
*(S
3)k(S
3)
k6 and the quantum double D(S
3)
\triangleleft $$
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k
S
3, finding respectively a natural calculus and a unique calculus with H
0 = k.1. 相似文献
5.
Axel Schüler 《Compositio Mathematica》2001,126(1):57-77
Let be one of the N
2-dimensional bicovariant first-order differential calculi on the quantum groups O
q
(N) or Sp
q
(N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra
s
is the quotient of the universal exterior algebra
u
by the principal ideal generated by . Here denotes the unique up to scalars bi-invariant 1-form. Moreover, is central in
u
and
u
is an inner differential calculus. 相似文献
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