Galois objects for algebraic quantum groups

Abstract

The basic elements of Galois theory for algebraic quantum groups were given in the paper 'Galois Theory for Multiplier Hopf Algebras with Integrals' by Van Daele and Zhang. In this paper, we supplement their results in the special case of Galois objects: algebras equipped with a Galois coaction by an algebraic quantum group, such that only the scalars are coinvariants. We show how the structure of these objects is as rich as the one of the quantum groups themselves: there are two distinguished weak K.M.S. functionals, related by a modular element, and there is an analogue of the antipode squared. We show how to reflect the quantum group across a Galois object to obtain a (possibly) new algebraic quantum group. We end by considering an example.

Comonoidal W*-Morita equivalence for von Neumann bialgebras

“…In case the corners of Q are Hopf algebras, this weak comonoidal structure can be shown to be strong. A similar discussion then holds in the analytic setting: for a general linking weak von Neumann bialgebra, we will get a weakly comonoidal -equivalence between the monoidal categories Rep of normal unital -representations of the corner von Neumann algebras on Hilbert spaces, and this will be strongly comonoidal if these corners are von Neumann algebraic quantum groups (see again [6] for details). In any case, we have seen that it is the comonoidal structure which appears most naturally, hence we use it to designate the structure.…”

“…Indeed, there a notion of Galois objects was introduced. Although one can in fact obtain a complete duality theory between Galois objects (for a von Neumann algebraic quantum group) and Galois co-objects (for the dual von Neumann algebraic quantum group), we have refrained from carrying out this discussion in full here, as the details are somewhat technical (in essence, the details of the duality construction can be found in [6], but one first needs to prove Theorem 0.7 of the present paper to be able to use those results).…”

“…As linking von Neumann bialgebras between von Neumann algebraic quantum groups turn out to have a lot of extra structure, such as an associated C -algebraic description (see again [6]), we prefer to use the following terminology in this case.…”

I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type

“…For ω, χ two 2-cocycle functionals on H, we define ω H χ −1 to be the vector space H with the new product h · g = ω(h (1) , g (1) )h (2) g (2) χ −1 (h (3) , g (3) ).…”

On projective representations for compact quantum groups

“…In case the irreducible (N, N )-corepresentations are finite-dimensional, the linear span of the W * r,ij generates inside N a purely algebraic Galois co-object N for the Hopf algebra A inside (M, M ). Conversely, if one starts with a Galois co-object for A , satisfying some suitable relations with the * -structure, we can in essence develop the whole theory so far in an algebraic way, and then necessarily the reflection will correspond to a compact quantum group (this was essentially already observed in [7]). As it turns out, there do exist interesting Galois co-objects which are of a non-algebraic type (see the final section), which was part of the motivation for writing this paper.…”