A Q-operator for open spin chains I: Baxter's TQ relation

We construct a Q-operator for the open XXZ Heisenberg quantum spin chain with diagonal boundary conditions and give a rigorous derivation of Baxter’s TQ relation. Key roles in the theory are played by a particular infinite-dimensional solution of the reflection equation and by short exact sequences of intertwiners of the standard Borel subalgebras of $U_q(\widehat{{\mathfrak{s}\mathfrak{l}}_2})$. The resulting Bethe equations are the same as those arising from Sklyanin’s algebraic Bethe ansatz.