Recent Advances in Algebraic Bethe Ansatz

Recent development of the algebraic Bethe ansatz for the quantum integrable models associated with quantum affine algebras (QAA) and Yangian doubles is reviewed. New approach uses expressions for the off-shell Bethe vectors (BV) in terms of Cartan–Weyl generators of these algebras and allows to obtain effective formulas of the action of monodromy matrix entries onto BV and recurrence relations for them. Relations between Gauss coordinates of the fundamental L-operators and currents for classical series Yangians and QAAs are presented.