Representations of $\mathfrak{sl}_2(\mathbb k)$
The representation category of $\mathfrak{sl}_2(\mathbb k)$ has countably infinte simple objects $\{V_i \mid i \in \mathbb N_0\}$ where the tensor product is given by the Clebsch-Gordon rule
\[V_i \otimes V_j = ⨁\limits_{l = 0}^{\min{(i,j)}} V_{i+j - 2l}.\]
We can construct the category and specify at any value for $q$ that is either 1 or not a rot of unity.
TensorCategories.sl2_representations — Functionsl2_representations(F::Ring)
sl2_representations(F::Ring, q::RingElem)Construct a skeletal category equivalent to the category of representations of $𝔰𝔩₂(F)$ specialized at $q$. $q$ defaults to $1$.
Verlinde type categories
When specifying at a root of unity we arrive at the Verlinde category. These categoryies have $n$ simple objects $V_0,...,V_{n-1}$and fusion rule
\[V_i \otimes V_j = ⨁\limits_{l = 0}^{\min{(i,j)}} V_{i+j - 2l}.\]