# Fusion ring example

## example

• $\mathfrak{g}=\mathfrak{sl}_2$
• let $k=3$ be the level for the fusion ring $C_{\mathfrak{g},k}$
• $\operatorname{Rep}(\mathfrak{g})$ has basis $\{V_0,V_1,V_2,V_3,V_4,\cdots \}$ with product

$$V_{m}{\otimes}V_{n}\cong V_{|m-n|}\oplus V_{|m-n|+2}\oplus \cdots \oplus V_{m+n}$$

• $C_{\mathfrak{g},k}$ has basis $\{V_0,V_1,V_2,V_3\}$ with product

$$V_{m}{\otimes}V_{n}\cong V_{|m-n|}\oplus V_{|m-n|+2}\oplus \cdots \oplus V_{\operatorname{min}(2k-(m+n),m+n)}$$

• $(V_1)^{\otimes l}$ for $l\geq 0$ in each ring :

$$\begin{array}{c|c|c} l & \operatorname{Rep}(\mathfrak{g}) & C_{\mathfrak{g},k} \\ \hline 0 & V_0 & V_0 \\ 1 & V_1 & V_1 \\ 2 & V_0\oplus V_2 & V_0\oplus V_2 \\ 3 & 2 V_1\oplus V_3 & 2 V_1\oplus V_3 \\ 4 & 2 V_0\oplus 3 V_2\oplus V_4 & 2 V_0\oplus 3 V_2 \\ 5 & 5 V_1\oplus 4 V_3\oplus V_5 & 5 V_1\oplus 3 V_3 \\ 6 & 5 V_0\oplus 9 V_2\oplus 5 V_4\oplus V_6 & 5 V_0\oplus 8 V_2 \\ \end{array}$$