Coherent States for Hopf Algebras

Abstract

Abstract. Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.

Every Quantum Minor Generates an Ore Set

“…In his thesis, the present author has developed a direct localization approach ( [15,17]) to the construction of the coset spaces of the quantum linear groups and the locally trivial quantum principal fibrations deforming the classical fibrations of the type G → G/B and having Hopf algebras as the replacements of the structure groups. Apart from the sketch in [15], the main part of that work has been still unpublished (however, a nontrivial application to quantum group coherent states and appropriate measure is exhibited in [16]). The present paper fills a part of this gap in view of the observation that the sets S w in SL q (n) are sets multiplicatively generated by a specific set of quantum minors attached to the permutation matrix w, namely the set of all principal (=lower right corner) quantum minors of the row-permuted matrix of generators T = (t…”

Non-Commutative Localization in Algebra and Topology

“…The compatibility above ensures that the localization lifts to a localization of (E −Mod) T ([122], 8.5). Hence, T is compatible with the localization in the usual sense, with numerous applications of this type of situation ( [118,120,121,122]). …”

Čech cocycles for quantum principal bundles

“…Though the base looks like CP 1 at the local algebra level, its further structures are nonclassical (e.g. the measure utilized in [16]).…”