Pseudo-Riemannian Lie algebras with coisotropic ideals and integrating the Laplace-Beltrami equation on Lie groups

TL;DR

This paper identifies a class of pseudo-Riemannian Lie groups where the Laplace-Beltrami equation simplifies to a first-order PDE, allowing explicit solutions and revealing nonlocal symmetries through a noncommutative integration approach.

Contribution

It introduces a new method for solving the Laplace-Beltrami equation on specific Lie groups using noncommutative integration and Fourier transforms, leading to explicit solutions and nonlocal symmetries.

Findings

Reduction of Laplace-Beltrami to first-order PDEs on certain Lie groups

Explicit solutions for examples including Heisenberg and non-unimodular groups

Discovery of nonlocal symmetry operators for the original equations

Abstract

We identify a class of left-invariant pseudo-Riemannian metrics on Lie groups for which the Laplace-Beltrami equation reduces to a first-order PDE and admits exact solutions. The defining condition is the existence of a commutative ideal $\mathfrak{h}$ in the Lie algebra $\mathfrak{g}$ whose orthogonal complement satisfies $\mathfrak{h}^\perp\subseteq\mathfrak{h}$. Using the noncommutative integration method based on the orbit method and generalized Fourier transforms, we reduce the Laplace--Beltrami equation to a first-order linear PDE, which can then be integrated explicitly. The symmetry of the reduced equation gives rise, via the inverse transform, to nonlocal symmetry operators for the original equation. These operators are generically integro-differential, contrasting with the polynomial symmetries appearing in previously studied classes. The method is illustrated by two examples: the Heisenberg group $\mathrm{H}_3(\mathbb{R})$ with a Lorentzian metric and a four-dimensional non-unimodular group with a metric of signature $(2,2)$. In the latter, classical separation of variables is not directly applicable, yet the noncommutative approach yields explicit solutions and reveals the predicted nonlocal symmetry.