derivative of the transfer matrix w.r.t the spectral parameter generates the continuous time Markov matrix of the exclusion process (or respectively the Hamiltonian of the quantum spin chain). The higher order logarithmic derivatives generates other local charges which commute with the Markov matrix (respectively with the Hamiltonian) because of the commutation relation .

In the next subsection \[subsec:parallel\_update\], we will use the inhomogeneous transfer matrix with very particular inhomogeneity parameters (different from $1$) to define a discrete time Markov matrix.

In the case of an spectral parameter, the inhomogeneous transfer matrix reads $$t(z|\mathbf{z})=tr_0\left( R_{0L}\left(z-z_L\right) \dots R_{02}\left(z-z_2\right)R_{01}\left(z-z_1\right) \right).$$

#### Open boundary conditions.

In the case of a model with open boundary conditions, the building blocks of integrable models (together with the $R$-matrix already introduced in the last paragraph) are the reflection matrices $K(z)$ and $\overline{K}(z)$, both depending on the spectral parameter $z$ and acting on a single copy $\cV$ of the tensor space (*i.e* on a single site of the lattice). The matrix $K(z)$ is associated to the left boundary and satisfies the reflection equation $$\label{eq:reflection_equation}
  R_{12}\left(\frac{z_1}{z_2}\right) K_1(z_1) R_{21}(z_1 z_2) K_2(z_2) =
  K_2(z_2) R_{12}(z_1 z_2) K_1(z_1) R_{21} \left(\frac{z_1}{z_2}\right).$$ The previous equation holds in $\cV\otimes \cV$. Once again the subscripts indicate on 