Non-Commutative Harmonic and Subharmonic Polynomials

TL;DR

This paper defines a noncommutative Laplace operator for polynomials, classifies symmetric harmonic and subharmonic polynomials in two variables, and explores their limited variety, aiming to inspire further PDE research in noncommutative settings.

Contribution

It introduces a noncommutative Laplace operator and classifies symmetric harmonic and subharmonic polynomials in two variables, revealing their constrained structure.

Findings

Limited dimension of subharmonic polynomials of degree > 4

Complete classification of harmonic polynomials in two variables

Potential for new PDE analysis methods in noncommutative algebra

Abstract

The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x=(x_1,...,x_g). The Laplacian Lap[p,h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p,h]=h^2\Delta_x p where \Delta_x p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p,h]=0 and subharmonic if the polynomial q(x,h):=Lap[p,h] takes positive semidefinite matrix values whenever matrices X_1,..., X_g, H are substituted for the variables x_1,...,x_g, h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial differential equations and inequalities.