Extensions of locally compact quantum groups and the bicrossed product construction

By S. Vaes, L. Vainerman. 2001

162 citations recorded by the source corpus.

Abstract In the framework of locally compact quantum groups, we study cocycle actions. We develop the cocycle bicrossed product construction, starting from a matched pair of locally compact quantum groups. We define exact sequences and establish a one-to-one correspondence between cocycle bicrossed products and cleft extensions. In this way, we obtain new examples of locally compact quantum groups.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/951556ed101b6ee5b9f7f8a26ff2f8abeb9a4c07

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In Algebra and Number Theory

1. K. De Commer
2. Martijn Caspers
3. Saad Baaj
4. S. Vaes
5. P. deSmedt
6. J. Quaegebeur
7. G. Skandalis
8. Moritz Weber

In Geometry and Topology

1. Martijn Caspers
2. K. De Commer
3. Sven Raum
4. Saad Baaj
5. Moritz Weber
6. S. Vaes
7. P. deSmedt
8. J. Quaegebeur

In Mathematical Physics

1. Martijn Caspers
2. K. De Commer
3. Sven Raum
4. S. Vaes
5. Saad Baaj
6. Moritz Weber
7. P. deSmedt
8. J. Quaegebeur

Citations in the corpus

• Non-Semi-Regular Quantum Groups Coming from Number Theory (2002, methodology): “∞(G 1),δ 1) and (L∞(G 2),δ 2) in the sense of [2], Definition 8.1. Denoting by σthe flip map, it means that τσis a matching of (L∞(G 2),δ 2) and (L∞(G 1),δ op 1 ) with trivial cocycles, in the sense of [19], Definition 2.1: (τ⊗ι)(ι⊗τ)(δ 1 ⊗ι) = (ι⊗δ 1)τ and (ι⊗τ)(τ⊗ι)(ι⊗δ 2) = (δ 2 ⊗ι)τ. (3.1) Referrin”
• Amenability and the bicrossed product construction (2001, methodology): “In [15], one also defines bicrossed products of quantum groups, and one makes the link with short exact sequences of locally compact quantum groups, called extensions.”
• Easy quantum groups and quantum subgroups of a semi-direct product quantum group (2013, background): “It is a very simple special case of the bicrossed-product constructions described in [26, 28, 29, 43].”
• Strictly Outer Actions of Groups and Quantum Groups (2002, unknown): “O A ] 2 3 M ay 2 00”
• Galois objects and cocycle twisting for locally compact quantum groups (2008, background): “15 of [25], (ω′ ⊗ ι)(ŴΩ∗) ∈ NφN , (ω ′ ⊗ ι)(Ŵ ) ∈ Nφ and ΛN ((ω ′ ⊗ ι)(ŴΩ)) = Λ((ω ⊗ ι)(Ŵ )).”
• Locally compact quantum groups (2017, background): “By crossed product duality [34] there exists a dual coaction α : N→ N⊗M of (M,∆) on N that is determined by α((ω ⊗ ι)(ŴΩ∗)) = (ω ⊗ ι⊗ ι)(Ŵ13Ŵ12Ω12), with ω ∈ M̂∗.”
• DOUBLE CROSSED PRODUCTS OF LOCALLY COMPACT QUANTUM GROUPS (2002, background): “In [16], it is proven that we can define a l.”