Communications in Mathematical Physics ( IF 2.102 ) Pub Date : 2020-05-23 , DOI: 10.1007/s00220-020-03729-w
Andreas Næs Aaserud, David E. Evans

We associate to each Temperley–Lieb–Jones C*-tensor category $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ with parameter $$\delta$$ in the discrete range $$\{2\cos (\pi /(k+2)):\,k=1,2,\ldots \}\cup \{2\}$$ a certain C*-algebra $${\mathcal {B}}$$ of compact operators. We use the unitary braiding on $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ to equip the category $$\mathrm {Mod}_{{\mathcal {B}}}$$ of (right) Hilbert $${\mathcal {B}}$$-modules with the structure of a braided C*-tensor category. We show that $${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$ is equivalent, as a braided C*-tensor category, to the full subcategory $$\mathrm {Mod}_{{\mathcal {B}}}^f$$ of $$\mathrm {Mod}_{{\mathcal {B}}}$$ whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories.

down
wechat
bug