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Given $B(\lambda)=T^0_1 (\lambda)$ the component of the monodromy matrix T that creates a state, $\lambda$ the spectral parameter and $| \Omega \rangle$ the reference ground state,

In "Quantum Groups in two dimensional physics"(Gomez-Ruiz Altaba- Sierra), trying to write the Algebraic Bethe Ansatz State $\Psi$, it first says that if the reference state $| \Omega \rangle$ is formed by more than one spin state, I need to use the creation operator given by its coproduct, i.e. $\Delta B(\lambda)$ if I have two spins states. I then would say that if I have N spins states, the operator that I need use is $\Delta^{(N-1)}B(\lambda)$. Then it defines the Algebraic Bethe Ansatz State as: $\Psi = \prod^M_{n=1} B(\lambda_n)|\Omega \rangle$. I understand that the problem is that: 1. I need a set of spectral parameters $\{ \lambda_n \}$ 2. I need unique parameters. I don't understand:

  1. Is it using the N-1 coproduct to create that state?
  2. and if so, why is it not using a sum over n spin flipped rather than a product? A product doesn't make sense to me not even mathematically.
    1. If it is NOT using the coproduct, when do I use the coproduct to obtain the right operator?? Thank you
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The coproduct increases the number of sites $N$ (the length of the chain), while the product of different $B$s increases the number of excitations (magnons) $M$. So each $B(\lambda_n)\equiv B^{(N)}(\lambda_n)=\Delta^{(N-1)} B(\lambda_n)$ creates one excitation with quasimomentum $p(\lambda_n)$. The coproduct does not change the parameter $\lambda_n$.

Graphically: as in that book draw $B$ by a horizontal line (auxiliary space) to which we attach $\lambda$, with one vertical line (local physical space) for each $N$, so that each crossing is an $R$-matrix, and fix the values of the spins on the endpoints of the horizontal line to get $B=T^0_1$ out of the monodromy matrix $T$. Then the coproduct increases the number of vertical lines by one, while taking a product of such $B$s for given $N$ corresponds adding a horizontal line with a new spectral parameter. In other words, the product corresponds to putting diagrams of $B$s on top of each other, each of with has its own auxiliary spectral parameter $\lambda_n$, which represents the composition of operators, corresponding to taking their matrix products.

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