Lagrangian multiform structure for the lattice Gel'fand–Dikii hierarchy

The lattice Gel'fand--Dikii hierarchy was introduced by Nijhoff et al (1992 Inverse Problems 8 597--621) as the family of partial difference equations generalizing to higher rank the lattice Korteweg--de Vries systems, and includes in particular the lattice Boussinesq system. We present a Lagrangian for the generic member of the lattice Gel'fand--Dikii hierarchy, and show that it can be considered as a Lagrangian 2-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus the multiform structure proposed by Lobb and Nijhoff (2009 J. Phys. A: Math. Theor. 42 454013) is extended to a multi-component system.