Duality for action bialgebroids
Sophie Chemla, Fabio Gavarini, Niels Kowalzig
TL;DR
The paper proves that linear duality and Drinfeld duality functors commute with the construction of action bialgebroids and their quantum analogues. By establishing categorical equivalences for Yetter–Drinfeld modules and comodules, it shows that braided commutative YD algebras remain so under duality, enabling a precise identification between duals of action bialgebroids and smash products with duals. In the quantum setting, Drinfeld functors are shown to preserve the action bialgebroid structure, yielding canonical isomorphisms such as R_h#F_h^ ext{vee} ≅ (R_h#F_h)^ ext{vee} and R_h#U_h' ≅ (R_h#U_h)'; these results connect quantisations of action groupoids with the quantum duality principle, and extend to linear duality beyond projectivity via topological/ind-projective techniques. Collectively, the work unifies classical and quantum dualities in the realm of action bialgebroids and quantum groupoids, with implications for the quantisation of action groupoids and their symmetries.
Abstract
We study the effect of linear duality on action bialgebroids (also known as smash product or scalar extension bialgebroids) and, for those bearing a quantisation nature, the effect of Drinfeld functors underlying the quantum duality principle. By means of various categorical equivalences, it is shown that any braided commutative Yetter-Drinfeld algebra over any bialgebroid is also a braided commutative Yetter-Drinfeld algebra over the respective dual bialgebroid. This implies that the action bialgebroid of the dual exists, which is then proven to be isomorphic, as a bialgebroid, to the dual of the initial action bialgebroid: in short, (linear) duality commutes with the action bialgebroid construction. Similarly, for quantum groupoids to which the Drinfeld duality functors apply and the quantum duality principle holds, these Drinfeld duality functors are shown to commute with the action bialgebroid construction as well.