OCR Output Comparison | Datalab (Chandra & Surya)

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3. To define the associator, notice that for $V,W,U∈\mathrm{Rep}(A)$ , both the spaces $(V\overline{\otimes}W)\overline{\otimes}U$ and $V\overline{\otimes}(W\overline{\otimes}U)$ are given by the retract of the idempotent

V\otimes W\otimes U\longrightarrow V\otimes W\otimes U, v\otimes w\otimes u\longmapsto 1_{(1)}.v\otimes 1_{(2)}.w\otimes 1_{(3)}.u,

which can be verified using ( Axiom 1 ). We then define the associator $a_{V,W,U}$ as the canonical map between the two retracts. One can check that $a_{V,W,U}$ is an $A$ -module map, and satisfies the pentagon equation.

4. Given $V\in\mathrm{Rep}(A)$ , we set the left unitor $l_V:A^l\overline{\otimes}V\longrightarrow V$ to be the restriction of the map

A^l\otimes V\longrightarrow V, x\otimes v\longmapsto x.v

on $A^l\overline{\otimes}V$ ; we set the right unitor $r_V:V\overline{\otimes}A^l\longrightarrow V$ to be the restriction of the map

V\otimes A^l\longrightarrow V, v\otimes y\longmapsto\varepsilon^{rr}(y).v

on $V\overline{\otimes}A^l$ . One can check that $l_V$ and $r_V$ are indeed invertible $A$ -module maps, and satisfy the triangle equations.

5. The left dual $V^L$ of an object $V\in\mathrm{Rep}(A)$ is given by the dual vector space $V^*:=\mathrm{Hom}(V,k)$ endowed with the $A$ -action

x.\omega=\omega(S(x).{-}), \forall\omega\in V^*, x\in A.

Similarly, the right dual $V^R$ of $V$ is given by $V^*$ endowed with $A$ -action

x.\omega=\omega(S^{-1}(x).{-}), \forall\omega\in V^*, x\in A.

Secondly, note that $\mathrm{Rep}(A)$ is clearly a finite $k$ -linear category, with the tensor product $\overline{\otimes}$ being bi- $k$ -linear. This concludes our construction of $\mathrm{Rep}(A)$ .

1.20 Example. We illustrate the above construction of $\mathrm{Rep}(A)$ when $A$ is the weak Hopf algebra $B\otimes B^{\mathrm{op}}$ defined in Example 1.17. Since a left $A$ -module is precisely a $B$ - $B$ -bimodule, $\mathrm{Rep}(A)$ is equivalent to $\mathrm{BiMod}(B|B)$ as categories. It remains to find the monoidal structure on $\mathrm{Rep}(A)$ . Given left $A$ -modules $V$ and $W$ , which we identify as $B$ - $B$ -bimodules, the underlying vector space of $V\overline{\otimes}W$ is the retract of the idempotent

V\otimes W\longrightarrow V\otimes W, v\otimes w\longmapsto v.p^{(1)}\otimes p^{(2)}.w.

The action of $a\otimes b\in B\otimes B^{\mathrm{op}}$ on $V\overline{\otimes}W$ is given by the restriction of the map

V\otimes W\longrightarrow V\otimes W, v\otimes w\longmapsto a.v.p^{(1)}\otimes p^{(2)}.w.b

on $V\overline{\otimes}W$ . Using Corollary 1.5, it can be shown that the $B$ - $B$ -bimodule $V\overline{\otimes}W$ is precisely $V\otimes_B W$ .

To find the tensor unit of $\mathrm{Rep}(B\otimes B^{\mathrm{op}})$ , one first computes

\varepsilon^{lr}:B\otimes B^{\mathrm{op}}\longrightarrow B\otimes B^{\mathrm{op}}, a\otimes b\longmapsto ab\otimes 1;

\varepsilon^{rr}:B\otimes B^{\mathrm{op}}\longrightarrow B\otimes B^{\mathrm{op}}, a\otimes b\longmapsto 1\otimes ab.