ScalingStacks

There is a “left” $2$-representation on ${\mathcal{U}}$

$$\Upsilon^{-}:{\mathcal{U}}\to\operatorname{End}\nolimits({\mathcal{U}}),\ e^{n}\mapsto e^{n}\otimes-$$

and a “right” $2$-representation on ${\mathcal{U}}$

$$\Upsilon^{+}:{\mathcal{U}}\xrightarrow[\sim]{\mathrm{rev}}{\mathcal{U}}^{\mathrm{rev}}\xrightarrow{e^{n}\mapsto-\otimes e^{n}}\operatorname{End}\nolimits({\mathcal{U}}).$$

The bimodule $2$-representation $L^{\pm}$ associated to $\Upsilon^{\pm}$ is given by

$$L^{\pm}(e^{r},e^{s},e^{n})=\delta_{s,r+n}L^{\pm}(r,n)$$

and it is left and right finite. Its left dual is isomorphic to the bimodule $2$-representation $R^{\pm}$ given by

$$R^{\pm}(e^{s},e^{r},e^{n})=\delta_{s,r+n}R^{\pm}(r,n)$$

while its right dual is isomorphic to $R^{\mp}\langle-\frac{1}{2}n(2r+n-1)\rangle$ (note that the action of ${\mathcal{U}}$ on the duals is obtained from the natural action of ${\mathcal{U}}^{\mathrm{rev}{\operatorname{opp}\nolimits}}$ by applying the isomorphism $\mathrm{rev}\circ{\operatorname{opp}\nolimits}$).