A quantum algebra approach to discrete equations on uniform lattices * ) F.J. Herranz, A. Ballesteros Departamento de F´ ısica, Universidad de Burgos, E-09001 Burgos, Spain J. Negro, L.M. Nieto Departamento de F´ ısica Te´orica, Universidad de Valladolid, E-47011 Valladolid, Spain Received 19 February 2001 A quantum algebra method for deducing the symmetries of discrete equations on uniform lattices is proposed. In principle, such a procedure can be applied to discretizations in a single coordinate (space or time) and the symmetries obtained in this way are indeed differential-difference operators. Firstly, the method is illustrated on two known examples that have been also analysed from the usual Lie symmetry approach: a uniform space lattice discretization of the (1+1) free heat-Schr¨odinger equation associated to a quantum Schr¨ odinger algebra, and a discrete space (1 + 1) wave equation provided by a quantum so(2, 2) algebra. Furthermore, we construct a discrete space (2 + 1) wave equation from a new quantum so(3, 2) algebra, to show that this method is useful in higher dimensions. Time discretizations are also commented. 1 Introduction Recently, the standard methods used in the analysis of differential equations have been adapted to the study of symmetries of difference and differential-difference equations in [1, 2]. From a different perspective, several discretizations of classical linear differential equations on (non-uniform) q-lattices have been constructed in [3–5], where it has been found that their symmetries verify q-deformed commu- tation relations with respect to the Lie algebra structure of the continuous sym- metries. However, Hopf algebra structures underlying these q-symmetry algebras have been not found yet. On the other hand, when the discretization of linear equations is constructed on uniform lattices it has been proven (at least for the (1 + 1) heat-Schr¨ odinger and wave equations) that the associated symmetries close the (non-deformed) Lie algebra structure [6, 7]. The aim of this contribution is to present a different and new approach to the study of discrete equations on uniform lattices through quantum algebras. The structure of the paper is as follows. In the next section we give an overview of the results concerning the difference symmetries of the discrete heat-Schr¨odinger and wave equations on the uniform (1 + 1) spacetime lattices that have been ob- tained by means of the Lie symmetry theory in [6, 7]. The main steps of the quantum algebra approach that we propose are summarized in the Section 3 together with two examples: the (1 + 1) heat-Schr¨ odinger and wave equations on a uniform space * ) Presented by F.J. Herranz at the DI-CRM Workshop held in Prague, 18–21 June 2000. Czechoslovak Journal of Physics, Vol. 51 (2001), No. 4 321