matrix $K(z)$ but similar ones hold also for the matrix $\overline{K}(z)$):

-   the reflection equation reads $$\label{eq:reflection_equation_additive}
    R_{12}\left(z_1-z_2\right) K_1(z_1) R_{21}(z_1+z_2) K_2(z_2) =
      K_2(z_2) R_{12}(z_1+z_2) K_1(z_1) R_{21} \left(z_1-z_2\right)$$

-   the Markovian property reads ${{\langle\sigma|}} K(z) = {{\langle\sigma|}}$

-   the regularity property reads $K(0)=1$

-   the unitarity property reads $K(z)K(-z) = 1$

We are now in position to define the inhomogeneous transfer matrix for a model with open boundary conditions [@sklyanin] $$\label{eq:inhomogeneous_transfer_matrix_open}
  t(z|\mathbf{z})=tr_0 \left(\widetilde{K}_0(z) R_{0,L}\left(\frac{z}{z_L}\right) \dots R_{0,1}\left(\frac{z}{z_1}\right)
  K_0(z) R_{1,0}(z z_1) \dots R_{L,0}(z z_L) \right),$$ where the dual boundary matrix $\widetilde{K}(z)$ is defined as[^5] $$\label{eq:Ktilde_from_Kb}
  \widetilde K_1(z)= tr_0\left(\overline{K}_0\left(\frac{1}{z}\right)\left(\left(R_{0,1}(z^2)^{t_1}\right)^{-1}\right)^{t_1}P_{0,1}\right),$$ or equivalently $$\label{eq:Kb_from_Ktilde}
  \overline{K}_1(z)= tr_0\left( \widetilde{K}_0\left(\frac{1}{z}\right)R_{01}\left(\frac{1}{z^2}\right)P_{01} \right).$$

The matrix $\widetilde{K}(z)$ satisfies the dual reflection equation $$\label{eq:reflection_equation_dual}
   \widetilde{K}_2(z_2)\,\left(R_{21}^{t_1}(z_1z_2)^{-1}\right)^{t_1}\,\widetilde{K}_1(z_1)\,R_{21}\left(\frac{z_2}{z_1}\right)
   =R_{12}\left(\frac{z_2}{z_1}\right)\,\widetilde{K}_1(z_1)\,\left(R_{12}^{t_2}(z_1z_2)^{-1}\right)^{t_2}\,\widetilde{K}_2(z_2).$$

Thanks to the Yang-Baxter equation and to the reflection equations and (or equivalently ), it is possible to show [@sklyanin] that the transfer matrix commutes for different values of the spectral parameter $$\label{eq:transfer_matrix_open_commutation}
 [t(x|\mathbf{z}),t(y|\mathbf{z})]=0.$$

In the context of continuous time exclusion processes or of quantum spin chains, the transfer matrix is often used with inhomogeneity parameters $z_i=1$. In this case the derivative with respect to the spectral parameter gives a continuous time Markov matrix (respectively a quantum spin chain Hamiltonian) with two-site local updating rules (respectively with 