Extended double-complex linear systems and new multiple infinite-dimensional hidden symmetries for the general symplectic gravity models

Abstract

By using a so-called extended double (ED)-complex method, the previously found doubleness symmetry for each member of the class of general symplectic gravity models is further exploited and extended. A 2(n+1)×2(n+1) matrix double-complex H-potential is constructed for any non-negative integer n, and the motion equations in two dimensions are written in a double-complex formulation. A double-duality mapping is proposed and two pairs of ED-complex Hauser-Ernst-type linear systems [J. Math. Phys. 21, 1126 (1980)] are established. Based on these linear systems, explicit formulations of new multiple hidden symmetry transformations for the studied theories are given. For any fixed n, these symmetry transformations are verified to constitute multiple infinite-dimensional Lie algebras, each of which is a semidirect product of the Kac-Moody sp(2(n+1̂),R) and Virasoro algebras (without center charges). These results demonstrate that the ED-complex method is necessary and more effective, and the general symplectic gravity models under consideration possess much richer symmetry structures than previously expected.