Lie algebra correspondence for nonlocal symmetries

Let $\mathcal{G}$ be the Lie group of nonlocal symmetry diffeomorphisms associated with the integrable map

$$, respectively$$

. Let its Lie algebra of generators be the Lie algebra of the corresponding nonlocal symmetries. The continuous dynamical systems are denoted by

$$, respectively$$

, and their classical Lie point symmetries form Lie algebras. Lie algebra correspondence conjecture. The Lie algebra of the generators of the Lie group $\mathcal{G}$ of the nonlocal symmetry diffeomorphisms associated with the map

$$, respectively$$

, contains a subalgebra which is isomorphic to the Lie algebra of the classical Lie point symmetries of the continuous dynamical system

$$, respectively$$

. This is a weaker correspondence than an isomorphism of the full Lie algebras: it asserts only that the continuous system's classical symmetry algebra embeds as a subalgebra of the discrete system's nonlocal symmetry algebra. Whether this subalgebra exists remains open.