arXiv math/00022292000-02-27math.QA

Locally trivial quantum vector bundles and associated vector bundles

Dirk Calow, Rainer Matthes

备注：: 9 pages

英文摘要：

We define locally trivial quantum vector bundles (QVB) and QVB associated to locally trivial quantum principal fibre bundles. There exists a differential structure on the associated vector bundle coming from the differential structure on the principal bundle, which allows to define connections on the associated vector bundle associated to connections on the principal bundle.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A locally trivial quantum Hopf bundle

We describe a locally trivial quantum principal U(1)-bundle over the quantum space S^2_{pq} which is a noncommutative analogue of the usual Hopf bundle. We also provide results concerning the structure of its total space algebra (irreducible *-representations and topological K-groups) and its Galois aspects (Galois property, existence of a strong connection, non-cleftness).

2002-10-14math.QA

Connections on locally trivial quantum principal fibre bundles

Following the approach of Budzyński and Kondracki, we define covariant differential algebras and connections on locally trivial quantum principal fibre bundles. We also consider covariant derivatives, connection forms and curvatures and explore the relations between these notions. As an example, a U(1) quantum principal bundle over a glued quantum sphere and a connection in this bundle is constructed. This connection may be interpreted as a q-deformed Dirac monopole.

2000-02-27math.QA

Nontrivial Deformation of a Trivial Bundle

The ${\rm SU}(2)$-prolongation of the Hopf fibration $S^3\to S^2$ is a trivializable principal ${\rm SU}(2)$-bundle. We present a noncommutative deformation of this bundle to a quantum principal ${\rm SU}_q(2)$-bundle that is not trivializable. On the other hand, we show that the ${\rm SU}_q(2)$-bundle is piecewise trivializable with respect to the closed covering of $S^2$ by two hemispheres intersecting at the equator.

2014-03-28math.QA

Gauge transformations on locally trivial quantum principal fibre bundles

If P, B, H are the algebras of the total space, the base space, and the structure group of a locally trivial principal fibre bundle (QPFB), left (right) gauge transformations are defined as automorphisms of the left (right) B-module P which are adapted to the coaction of the Hopf algebra H and to the covering related to the local trivializations. Covariant derivatives on a QPFB are always transformed into covariant derivatives. This is true for connections only for special choices of the differential structure on H and/or if one restricts to algebra automorphisms. There are analogues of the cl…

2000-02-27math.QA

Reductions of piecewise-trivial principal comodule algebras

Let $G'$ be a closed subgroup of a topological group $G$. A principal $G$-bundle $X$ is reducible to a locally trivial principal $G'$-bundle $X'$ if and only if there exists a local trivialisation of $X$ such that all transition functions take values in $G'$. We prove a noncommutative-geometric counterpart of this theorem. To this end, we employ the concept of a piecewise-trivial principal comodule algebra as a replacement of a locally trivial compact principal bundle. To illustrate our theorem, first we define a new noncommutative deformation of the $\mathbb{Z}/2\mathbb{Z}$-principal bundle $…

2021-02-04math.QA

Knizhnik-Zamolodchikov bundles are topologically trivial

We prove that the vector bundles at the core of the Knizhnik-Zamolodchikov and quantum constructions of braid groups representations are topologically trivial bundles. We provide partial generalizations of this result to generalized braid groups. A crucial intermediate result is that the representation ring of the symmetric group on n letters is generated by the alternating powers of its natural n-dimensional representation.

2008-09-21math.ATmath.QA

更多相关论文收起更多论文

The local-triviality dimension of actions of compact quantum groups

We define the local-triviality dimension for actions of compact quantum groups on unital C*-algebras. The resulting compact quantum principal bundle is said to be locally trivial when this dimension is finite. For commutative C*-algebras, this notion recovers the standard definition of local triviality of compact principal bundles. We prove that actions with finite local-triviality dimension are automatically free. Then we apply this new notion to prove the noncommutative Borsuk-Ulam-type conjecture under the assumption that a compact quantum group admits a non-trivial classical subgroup whose…

2019-07-18math.OAmath.QA

A Locally Trivial Quantum Hopf Fibration

The irreducible *-representations of the polynomial algebra O(S^3_{pq}) of the quantum 3-sphere introduced by Calow and Matthes are classified. The K-groups of its universal C*-algebra are shown to coincide with their classical counterparts. The U(1)-action on O(S^3_{pq}) corresponding for p=1=q to the classical Hopf fibration is proven to be Galois (free). The thus obtained locally trivial Hopf-Galois extension is shown to be relatively projective (admitting a strong connection) and non-cleft. The latter is proven by determining an appropriate Chern-Connes pairing.

2002-09-26math.OAmath.QA