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} $$