Theoretical and Mathematical Physics

, Volume 196, Issue 3, pp 1307–1319 | Cite as

Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

  • D. LeviEmail author
  • L. Martina
  • P. Winternitz


The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E2. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane.


Lie group partial differential equation discretization procedure 


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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomeItaly
  2. 2.Instituto Nazionale di Fisica NucleareSezione di Roma TreRomeItaly
  3. 3.Dipartimento di Matematica e FisicaUniversità del SalentoLecceItaly
  4. 4.Instituto Nazionale di Fisica NucleareSezione di LecceLecceItaly
  5. 5.Département de Mathématiques et de Statistique and Centre de Recherches MathématiquesUniversité de MontréalMontréalCanada

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