nLab tube algebra

Definition

Let $\mathcal{C}$ be a fusion category. The vector space

$$Tube (\mathcal{C}) \coloneqq \bigoplus_{a,b,c \in \mathcal{I}} \mathcal{C}(c \otimes a, b \otimes c),$$

for $\mathcal{I}$ the set of simple objects of $\mathcal{C}$, has an algebra structure. This is called the Tube (or Ocneanu) algebra. For $\epsilon \in \mathcal{C}(x \otimes a, a \otimes y)$, $\epsilon' \in \mathcal{C}(x' \otimes a', a' \otimes y')$, their product is given by

$$\epsilon \cdot \epsilon' = \delta_{y,x'} \sum_{b,\alpha} (\alpha \otimes \text{id}_{y'}) (\text{id}_a \otimes \epsilon') (\epsilon \otimes \text{id}_{a'}) (\text{id}_x \otimes \alpha^* ),$$

for $\alpha \in \text{Hom}(a \otimes a', b)$ (Ocneanu 94). It is furthermore a C-star-algebra (Ocneanu 01), and even a weak Hopf algebra (Müger 03, Jia et al 24).

Properties

Relation to the Drinfeld center

The category of representations of the tube algebra $Tube(\mathcal{C})$ of $\mathcal{C}$ is equivalent to the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of $\mathcal{C}$ (Müger 03).

Relation to action on Hilbert spaces

In the context of a 2d CFT, twist fields are thought of as elements of a twist Hilbert space $\mathcal{H}_a$ for $a \in \mathcal{I}$. The action of $b\in \mathcal{I}$ (essentially by conjugation) is described by an element $x\in \mathcal{C}( b\otimes a, c \otimes b)$. Thus, the tube algebra $Tube(\mathcal{C})$ acts on the total Hilbert space $\mathcal{H} = \bigoplus_{a\in \mathcal{I}} \mathcal{H}_a$, meaning the latter is a representation of the tube algebra, and by the equivalence above is identified with an object in the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of $\mathcal{C}$. See e.g. (Lin et al. 23).

References

General:

• Adrian Ocneanu. Chirality for operator algebras. Subfactors (Kyuzeso, 1993) 39 (1994). (pdf).

• Adrian Ocneanu. Operator algebras, topology and subgroups of quantum symmetry–construction of subgroups of quantum groups–. Taniguchi Conference on Mathematics Nara’98. Vol. 31. Mathematical Society of Japan, 2001. (doi).

• Michael Müger. From subfactors to categories and topology II: The quantum double of tensor categories and subfactors. Journal of Pure and Applied Algebra 180.1-2 (2003): 159-219. (doi00248-7)).

On its structure

• Zhian Jia, Sheng Tan, and Dagomir Kaszlikowski. Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model. Journal of High Energy Physics 2024.7 (2024): 1-63. (doi207)).

As encoding actions on Hilbert spaces

• Ying-Hsuan Lin, Masaki Okada, Sahand Seifnashri, and Yuji Tachikawa. Asymptotic density of states in 2d CFTs with non-invertible symmetries. Journal of High Energy Physics 2023, no. 3 (2023): 1-43. (doi094)).

• Yichul Choi, Brandon C. Rayhaun, Yunqin Zheng. Generalized Tube Algebras, Symmetry-Resolved Partition Functions, and Twisted Boundary States (2024). (arXiv:2409.02159).