Non-Commutative Harmonic and Subharmonic Polynomials
Abstract
The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x=(x1,...,xg). 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]=h2x p where 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 X1,..., Xg, H are substituted for the variables x1,...,xg, 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.