Spectral Fusion Deformations for Locally Compact Quantum Groups

TL;DR

This work introduces a purely algebraic deformation framework for $C^*$-algebras equipped with a coaction of a locally compact quantum group, defined at the level of finite spectral cores and their fusion structure. Deformations are governed by fusion data via channelwise phases $\omega(\pi,\sigma,\tau)$ that satisfy a fusion pentagon, ensuring associativity of the deformed product $\star_\omega$ on $A_{\mathrm{fin}}$ and extending to a Fréchet $*$-algebra $A_{\infty}$. A minimal reduced setting is identified where the deformed left regular action is bounded, yielding a canonical reduced $C^*$-completion $A_{\omega}$; the construction separates algebraic coherence from analytic implementability and yields genuinely new associator-driven deformations beyond dual $2$-cocycles. The framework recovers classical deformations (Moyal–Weyl, Rieffel, Kasprzak, Drinfeld twists, Connes–Landi) and includes new examples such as associator deformations of $C(\mathrm{SU}_q(3))$, fusion-graph algebras, and cyclic fusion deformations, illustrating that higher-order fusion data can produce nontrivial deformations invisible to $2$-cocycle methods. Overall, the work clarifies the role of fusion coherence in deformation theory for locally compact quantum groups and unifies multiple deformation mechanisms under a single spectral-fusion paradigm.

Abstract

We develop a deformation framework for $C^*$-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 $*$-algebra completion under mild analytic regularity assumptions. Deformations are governed by scalar fusion data assigning phases to fusion channels of irreducible corepresentations. Associativity and $*$-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 $C^*$-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.

Theorems & Definitions (14)

• Definition 2.1
• Definition 2.2: Finite spectral core
• Remark 2.3
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3: Reduced Hilbert--module representation
• proof
• Proposition 3.4: Uniform $L^2$--boundedness of left multiplication