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A particular example is the quantum enveloping algebra U_q(𝔰𝔩₂) acting on the quantum plane R=k⟨ x, y | xy = qyx ⟩.
Next up is Victor Ostrik with an introduction to fusion categories. In the passage from classical symmetries and quantum symmetries, finite groups are to groups as fusion categories are to (ℂ-)linear tensor categories.
The fundamental example for this talk is Vec_{ℤ/n}^ω. The objects are elements i,j etc of ℤ/n, and i⊗j ≔ i+j. The interesting part is the associators, which are specified by invertible elements ω(i,j,k) in the ground field.
For the pentagon axiom to hold, we need ∂ω=1 as a 4-cochain. As an example when n=2, we can take ω(i,j,k) = -1 when i=j=k=1, and ω(i,j,k)=1 otherwise.
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