TABLE OF CONTENTS In this chapter we shall describe the algebraic Bethe ansatz. This version of the Bethe ansatz will be very useful for following studies of inhomogeneous quantum chains, the main topic of this book. Generally speaking, it is very important for the search of models which permit Bethe ansatz solutions.
As we pointed out in the previous chapter, when we studied the nested Bethe ansatz for a Hubbard chain, the problem for a quantum spin- chain can be solved in a different way than using the co-ordinate Bethe ansatz from Chapter 3. Namely, we take into account that the condition on a Bethe ansatz two-spin nested wave function for a Hubbard chain (which is, in fact, the wave function of an inhomogeneous one-dimensional Heisenberg spin- model) can be written as
Here we used the wave function of a Hubbard chain
where x 1, , x M ? denotes the wave function with M down spins (of N electrons) at positions x 1 < < x M and the two-particle scattering matrix Y of a Hubbard chain.
In general for a SU(2)-symmetric spin- chain of length L one can in-troduce spectral parameters ? (which, e.g., can be related to momenta of
Bethe ansatz wave functions), so that
with 
and 
being the identity and permutation operators acting in the Hilbert space of the m-th and n-th spin...
