# Conformally invariant elliptic Liouville equation and its symmetry preserving discretization

## TL;DR

This paper explores the symmetry properties of the elliptic Liouville equation, revealing its conformal invariance, and develops a discretization scheme that preserves this symmetry, ensuring the discrete model closely aligns with the continuous equation.

## Contribution

It introduces a symmetry-preserving discretization scheme for the elliptic Liouville equation that maintains the conformal group invariance in the discrete setting.

## Key findings

- The discretization scheme is invariant under the group O(3,1).
- The scheme's continuous limit recovers the elliptic Liouville equation in polar coordinates.
- The lattice used in the scheme is invariant under O(2) rotations.

## Abstract

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 $E_2$. 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 discretisation procedure developed earlier, 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.