2002/04/23 by Cyril Grunspan, Grunspan, Cyril
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings, Modules, and Algebras #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0204280
12 pages
arxiv created 2002/04/23 · openalex publication_date 2002/04/23 · arxiv updated 2009/11/30 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
This text gives some results about quantum torsors. Our starting point is an old reformulation of torsors recalled recently by Kontsevich. We propose an unification of the definitions of torsors in algebraic geometry and in Poisson geometry. Any quantum torsor is equipped with two comodule-algebra structures over Hopf algebras and these structures commute with each other. In the finite dimensional case, these two Hopf algebras share the same finite dimension. We show that any Galois extension of a field is a torsor and that any torsor is a Hopf-Galois extension. We give also examples of non-commutative torsors without character. Torsors can be composed. This leads us to define a new group-invariant, its torsors invariant. We show how Parmentier's quantization formalism of "affine Poisson groups" is part of our theory of torsors.