Hopf-kernel characterization of confinement walls

Let $\mathcal{T}$ be the residual symmetry algebra, let $\mathcal{U}$ be the non-confined algebra, and let $\Gamma:\mathcal{T}\to\mathcal{U}$ be the Hopf map. Define the left and right Hopf kernels by

$${\rm LKer}(\Gamma):=\{t\in\mathcal{T}\mid(\Gamma\otimes {\rm id})\circ\Delta(t)=1_{\mathcal{U}}\otimes t\},$$

and

$${\rm RKer}(\Gamma):=\{t\in\mathcal{T}\mid({\rm id}\otimes\Gamma)\circ\Delta(t)=t\otimes 1_{\mathcal{U}}\}.$$

For a $\mathcal{T}$-irrep $\Omega^{B}_{\beta}$, the Hopf-kernel wall characterization. The wall corresponding to $\Omega^{B}_{\beta}$ is characterized by the restriction of $\Omega^{B}_{\beta}$ to either ${\rm LKer}(\Gamma)$ or ${\rm RKer}(\Gamma)$.

These two candidate subalgebras are respectively right and left coideal subalgebras, and their representations correspond under the antipode; under the assumed semisimplicity they are isomorphic. The statement identifies the wall with the corresponding restricted representation, but the supplied text does not establish it as a theorem.