I'm a mathematician who's trying to understand the meaning of Algebraic Bethe Ansatz. What I understood is that when dealing with quantum integrable models (like XXZ Heisenberg spin chain), one is interested in showing that there are conserved quantities (ie operators that commute with the Hamiltonian).

My questions are about the tools that we have to use in order to exhibit them. Can someone give me references or explain to me in which way I should think about these objects  ?

1. The Lax operator
2. The R matrix
3. The monodromy matrix
4. The transfer matrix

I understand that one defines a Lax operator then show that it verifies a certain commutation relation with the R matrix (which verifies Yang Baxter equation). One multiplies the Lax operators along all sites and obtain the monodromy matrix, taking its trace yields the transfer matrix. Finally one can show that this last operator can be seen as the generating function of conserved quantites. Do not hesitate to correct me where I'm wrong.

Thank you in advance.

I'm a mathematician who's trying to understand the meaning of Algebraic Bethe Ansatz. What I understood is that when dealing with quantum integrable models (like XXZ Heisenberg spin chain), one is interested in showing that there are conserved quantities (ie operators that commute with the Hamiltonian).

My questions are about the tools that we have to use in order to exhibit them. Can someone explain to me in which way I should think about these objects?

1. The Lax operator
2. The $R$ matrix
3. The monodromy matrix
4. The transfer matrix

I understand that one defines a Lax operator then show that it verifies a certain commutation relation with the $R$ matrix (which verifies Yang-Baxter equation). One multiplies the Lax operators along all sites and obtain the monodromy matrix, taking its trace yields the transfer matrix. Finally one can show that this last operator can be seen as the generating function of conserved quantites. Do not hesitate to correct me where I'm wrong.