The local-triviality dimension of actions of compact quantum groups

2018/01/02 by Eusebio Gardella, Gardella, Eusebio, Piotr M. Hajac +5
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1801.00767

openalex publication_date 2018/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 induced action has finite local-triviality dimension. This is a noncommutative extension of the Borsuk-Ulam-type theorem for locally trivial principal bundles.

Citations

• Noncommutative Geometry Approach to Principal and Associated Bundles
• Completely positive maps of order zero
• The weak Hilbert-Smith conjecture from a Borsuk-Ulam-type conjecture
• Equivariant semiprojectivity

Related

• Continuity and equivariant dimension
• The unitary implementation of a locally compact quantum group action
• Rokhlin dimension for compact quantum group actions
• Noncommutative numerable principal bundles from group actions on C*-algebras
• Property T for general locally compact quantum groups