2023/12/01 by Pierre Bieliavsky, Victor Gayral, Bieliavsky, Pierre +5 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2312.00523
openalex publication_date 2023/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given an extension 0→ V→ G→ Q→1 of locally compact groups, with V abelian, and a compatible essentially bijective 1-cocycle η\colon Q→ V, we define a dual unitary 2-cocycle on G and show that the associated deformation of G is a cocycle bicrossed product defined by a matched pair of subgroups of Q\ltimes V. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang--Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of G on L2(Q) and a unitary quantization map L2(G)→\rm HS(L2(Q)) of Kohn--Nirenberg type.