Spectral Fusion Deformations for Locally Compact Quantum Groups

Abstract

We develop a deformation framework for $C^{\ast }$-algebras equipped with a coaction of a locally compact quantum group, formulated intrinsically at the level of spectral subspaces determined by the coaction. The construction is defined algebraically on a finite spectral core and extended by continuity to a natural Fréchet $\ast$-algebra completion under mild analytic regularity assumptions. Deformations are governed by scalar fusion data assigning phases to fusion channels of irreducible corepresentations. Associativity and $\ast$-compatibility are characterized by explicit algebraic identities. The framework recovers a range of known deformation procedures, including Rieffel, Kasprzak, and Drinfeld-type constructions, and also yields genuinely new deformations that do not arise from dual $2$--cocycles or crossed-product methods. At the $C^{\ast }$-level, we identify a minimal reduced setting in which the deformed algebra admits a canonical completion, formulated in terms of boundedness of the deformed left regular action on the Haar--GNS space. This separates algebraic coherence from analytic implementability and clarifies the precise role of higher-order fusion data in deformation theory for locally compact quantum groups. In particular, the framework exhibits explicit associator-level deformations governed by fusion $3$--cocycles that cannot arise from any dual $2$--cocycle or crossed-product construction.

arXiv commentWithdrawn by the author due to (i) incorrect spectral assumptions: the algebraic spectral core defined via point isotypic components need not be dense outside discrete/Peter -- Weyl-type settings, and (ii) a flaw in the deformation mechanism whereby the proposed 3-cocycle deformation collapses to a "lazy" 2-cocycle and does not yield a genuine associativity deformation in the stated generality