A few remarks on the tube algebra of a monoidal category

Abstract

We prove two results on the tube algebras of rigid C${}^{\ast }$-tensor categories. The first is that the tube algebra of the representation category of a compact quantum group $G$ is a full corner of the Drinfeld double of $G$. As an application we obtain some information on the structure of the tube algebras of the Temperley-Lieb categories $TL(d)$ for $d>2$. The second result is that the tube algebras of weakly Morita equivalent C${}^{\ast }$-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the $2$-category defining the Morita context.

17 pages; v4: minor changes and change of section numbering to the published version, to appear in Proc. Edinb. Math. Soc.; v3: description of the topology on the primitive spectrum of the tube algebra of a Temperley-Lieb category (d>2), a few remarks and minor fixes; v2: rewritten last section

References in corpus (3)

• Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)
• Representation theory for subfactors, $\lambda$-lattices and C*-tensor categories
• Drinfeld center and representation theory for monoidal categories

Cited by in corpus (6)

• Dualities in one-dimensional quantum lattice models: topological sectors
• Low-depth unitary quantum circuits for dualities in one-dimensional quantum lattice models
• Q-systems and extensions of completely unitary vertex operator algebras
• Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs
• The approximation property for locally compact quantum groups
• From gauging to duality in one-dimensional quantum lattice models