非交换调和与次调和多项式
Non-Commutative Harmonic and Subharmonic Polynomials
摘要：
本文引入了多项式p在非交换变量x=（x_1，...，x_g）。p的Laplacian Lap[p，h]是关于x和非交换变量h的多项式。当所有变量都可交换时，我们有Lap[p，h]=h^2Δ_xp，其中Δ_xp是通常的拉普拉斯算子。如果Lap[p，h]=0，则称对称变量的对称多项式为调和多项式，如果多项式q（x，h）：=Lap[p，h]取正半定矩阵值，则称其为次调和多项式，只要矩阵X_1，.，X_g，H被变量x_1，.，x_g，h.本文对两个对称变量的齐次对称调和多项式和次调和多项式进行了分类。我们发现它们并不多：例如，所有高于4次的次谐波的跨度都有2维（如果是奇数次）和3维（如果是偶数次）。希望这里的方法能为定义和分析其他偏微分方程和不等式提供一些建议。
英文摘要：
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^2Δ_x p where Δ_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.
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