We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras
supd(2). These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the
supd(2) defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.
In this body of work, we will examine two classes of Lie algebra deformations and their appli- cation to the exact solutions of various quantum integrable systems. The first class of Lie algebra deformations are polynomial deformations of the A1 Lie algebra. We construct unitary representations of these algebras in terms of bosonic and spin operators as well as obtain their single-variable differential operator realizations. Using these results, we obtain the exact Bethe ansatz solutions of a variety of quantum integrable systems such as the Bose-Hubbard and Tavis-Cummings model. The second class of Lie algebra deformations are the so-called quasi-Gaudin algebra. This algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation this algebra, we obtain partial solutions for a class of bosonic models which do not preserve U(1) symmetry
A quantum integrable spin chain model associated with the G 2 exceptional Lie algebra is studied. By using the fusion technique, the closed recursive relations among the fused transfer matrices are obtained. These identities allow us to derive the exact energy spectrum and Bethe ansatz equations of the system based on polynomial analysis. The present method provides a unified treatment to investigate the Bethe ansatz solutions for both the periodic and the non-diagonal open boundary conditions associated with exceptional Lie algebras.