Quantum torsors

Abstract

The following text is a short version of a forthcoming preprint about torsors. The adopted viewpoint is an old reformulation of torsors recalled recently by Kontsevich [Kon]. We propose a unification of the definitions of torsors in algebraic geometry and in Poisson manifolds (Example 2 and section 2.2). We introduce the notion of a quantum torsor (Definition 2.1). Any quantum torsor is equipped with two comodule-algebra structures over Hopf algebras and these structures commute with each other (Theorem 3.1.) In the finite dimensional case, these two Hopf algebras share the same finite dimension (Proposition 3.1). We show that any Galois extension of a field is a torsor (Example 4) and that any torsor is a Hopf-Galois extension (section 3.2). We give also examples of non-commutative torsors without character (Example 5). Torsors can be composed (Theorem 3.2). This leads us to define for any Hopf algebra, a new group-invariant, its torsors invariant (Theorem 3.3). We show how Parmentier's quantization formalism of "affine Poisson groups" is part of our theory of torsors (Theorem 3.4).I am very grateful to B. Enriquez for his help and his support. I am also grateful to P. Cartier, A. Chambert-Loir, S. Natale and S. Parmentier for comments and remarks.

Galois objects for algebraic quantum groups

“…But for a Galois object, the antipode is really just a formal construction using its antipode squared. We want to remark that the notion of an 'antipode squared' on a Galois object for a Hopf algebra was considered more or less in [11], but in a different set-up. Also, the antipode squared there was a part of the axiom system.…”

On the Construction of Quantum Homogeneous Spaces from ∗ -Galois Objects

“…These co-linking weak Hopf- * -algebras can be seen as specializations of Takeuchi's pre-equivalences (or strict Morita-Takeuchi-contexts as they are now called). The notion of a co-linking weak Hopf * -algebra can also be shown to be equivalent with that of a total Hopf-Galois system of [11] (equipped with a * -structure), but using the language of weak Hopf algebras makes the definition somewhat more concise. The proof of the equivalence between these two concepts is essentially the one of Proposition 2.8.…”

Pre-torsors and Galois Comodules Over Mixed Distributive Laws

“…it contains only multiplication by k), is called an H-Galois object. As observed by Grunspan [9] and Schauenburg [14,15,17], faithfully flat Hopf-Galois objects can be described equivalently, without explicit mention of the coacting Hopf algebra H, in terms of torsors. A torsor means a certain map T → T ⊗ k T ⊗ k T, from which the flat Hopf algebra H can be reconstructed uniquely up to isomorphism.…”