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Let $\mathbb{A}$ be a Kac algebra (i.e. a Hopf C*-algebra) of finite dimension, and $\mathbb{A}^{*}$ its dual :

Is there a formula revealing the fusion rules for the irreducible representations of $\mathbb{A}$, by using the multiplication and the comultiplication, on $\mathbb{A}$ and $\mathbb{A}^{*}$ ?

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Definition : Let $\mathbb{A}$ be a finite dimensional Kac algebra, let $V$ and $W$ be two irreducible representations of $\mathbb{A}$ as C$^{*}$-algebra, then $\mathbb{A}$ acts on $V \otimes W$ by using the comultiplication $\Delta$:
$$\forall x \in \mathbb{A}, \forall v \in V, \forall w \in W : \Delta(x).(v\otimes w) = \sum (x_{(1)}.v)\otimes (x_{(2)}.w)$$

Remark : As a finite dimensional Kac algebra, $\mathbb{A}$ admits finitely many irreducible representations $H_{1}$, ..., $H_{r}$ of increasing dimension $n_{1} = 1$, $n_{2}$, ... , $n_{r}$.

Fusion rules : The previous action of $\mathbb{A}$ on $H_{i}\otimes H_{j}$ decomposes into irreducibles :
$$ H_{i}\otimes H_{j} = \bigoplus_{k}{M_{ij}^{k} \otimes H_{k}} $$
with $M_{ij}^{k}$ the multiplicity space of dimension $n_{ij}^{k}$, so that : $\sum n_{i}.n_{j} = \sum n_{ij}^{k} . n_{k}$

Schur's lemma : let ($\mathcal{A},V)$ be finite dimensional C$^{*}$-algebra and representation:

  • $\mathcal{A}$ acts irreducibly on $V$ (i.e has no invariant subspace) iff $\pi_{V}(\mathcal{A})' = \mathbb{C}I_{V}$
  • If $T \in Hom_{\mathcal{A}}(V_{1},V_{2})$ (i.e. commutes with $\mathcal{A}$) then $T=0$ or $T$ is an isomorphism.

Double commutant theorem: If $\mathcal{A} \subset End(V)$ is a C$^{*}$-subalgebra then : $\mathcal{A}'' = \mathcal{A}$

Corollary : As C$^{*}$-algebra, $\mathbb{A} \simeq \bigoplus_{i} M_{n_{i}}(\mathbb{C}) $.

Remark : $(H_{i}\otimes H_{j})_{i,j}$ are the irreducible representations of $\mathbb{A} \otimes \mathbb{A}$ :

  • $\pi_{H_{i}\otimes H_{j}}(\mathbb{A} \otimes \mathbb{A}) = \pi_{H_{i}\otimes H_{j}}(\mathbb{A} \otimes \mathbb{A})'' \simeq M_{n_{i}n_{j}}(\mathbb{C})$
  • Let $V = \bigoplus_{i,j} H_{i}\otimes H_{j}$ then $\pi_{V}(\mathbb{A} \otimes \mathbb{A}) = \pi_{V}(\mathbb{A} \otimes \mathbb{A})''\simeq \bigoplus_{i,j} M_{n_{i}n_{j}}(\mathbb{C}) $

Inclusion : The comultiplication $\Delta : \mathbb{A} \hookrightarrow \mathbb{A} \otimes \mathbb{A} $, gives an inclusion of C$^{*}$-algebra :

  • $\mathbb{A} \simeq \Delta(\mathbb{A}) \subset \mathbb{A}\otimes \mathbb{A}$
  • $[\Delta(\mathbb{A}) \subset \mathbb{A}\otimes \mathbb{A}]\simeq [\pi_{V}(\Delta(\mathbb{A})) \subset \pi_{V}(\mathbb{A}\otimes \mathbb{A})]$
  • $\pi_{H_{i}\otimes H_{j}}(\Delta(\mathbb{A})) = \pi_{H_{i}\otimes H_{j}}(\Delta(\mathbb{A}))'' \simeq \bigoplus_{k}{M_{ij}^{k} \otimes \pi_{H_{k}}(\mathbb{A})} \simeq \bigoplus_{k}{M_{ij}^{k} \otimes M_{n_{k}}(\mathbb{C})}$
  • $\pi_{H_{i}\otimes H_{j}}(\Delta(\mathbb{A})) \subset \pi_{H_{i}\otimes H_{j}}(\mathbb{A} \otimes \mathbb{A})$

Conclusion : The inclusion matrix of $[\Delta(\mathbb{A}) \subset \mathbb{A} \otimes \mathbb{A}]$ is $\Lambda = (\Lambda_{(i,j)}^{k})$ with $\Lambda_{(i,j)}^{k} = n_{ij}^{k}$, with $n_{ij}^{k} = dim(M_{ij}^{k})$. So the inclusion matrix (of the comultiplication) is given by the fusion rules.

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