Lie algebraic approach to the construction of (2+1)-dimensional lattice-field and field integrable Hamiltonian equations

Two different methods for the construction of (2+1)-dimensional integrable lattice-field and field Hamiltonian dynamical systems are presented. The first method is based on the so-called central extension procedure applied to the Lie algebra of shift operators and the Lie algebra of pseudodifferential operators. The second method is the so-called operand formalism. Both methods allow a construction of some new integrable nonlinear Hamiltonian lattice-field and field equations in (2+1)-dimensional space.