  The algebraic structure underlying the totally asymmetric exclusion process
is studied by using the Bethe Ansatz technique. From the properties of the
algebra generated by the local jump operators, we explicitly construct the
hierarchy of operators (called generalized hamiltonians) that commute with the
Markov operator. The transfer matrix, which is the generating function of these
operators, is shown to represent a discrete Markov process with long-range
jumps. We give a general combinatorial formula for the connected hamiltonians
obtained by taking the logarithm of the transfer matrix. This formula is proved
using a symbolic calculation program for the first ten connected operators.
  Keywords: ASEP, Algebraic Bethe Ansatz.
  Pacs numbers: 02.30.Ik, 02.50.-r, 75.10.Pq.
