$$ {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 $$-invariant non-compact boundary conditions for the XXZ spin chain

We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries chain, and $q=e^{\frac{i\pi}{p}}$ a $2p$-th root unity $ - its $p$-dimensional analogue. Both cases are parametrised by continuous "spin" $\alpha\in\mathbb{C}$. To motivate our construction, first specialise $q=i$, where obtain modified XX with unrolled quantum group symmetry, whose spectrum scaling limit is computed explicitly using free fermions. In continuum, this model identified $(\eta,\xi)$ ghost CFT upper-half plane continuum conformally invariant real axis. The different sectors irreducible Virasoro representations. Going back investigate algebraic properties underlying lattice algebras. show that if $q^\alpha\notin\pm q^{\mathbb{Z}}$, coupling provides faithful representation blob algebra which Schur-Weyl dual $U_q\mathfrak{sl}_2$. Then, modifying left right, universal two-boundary Temperley-Lieb algebra. generators parameters these representations in terms $\alpha$. Finally, conjecture general form duality case. This paper series will study, at all parameters, limit, content relevant algebras fusion chains.