Heat Semigroups on Weyl Algebra

TL;DR

This paper investigates the algebraic structure of heat semigroups generated by Laplacians on the Weyl algebra, providing Gaussian integral representations and explicit formulas for their products and heat kernels.

Contribution

It introduces a novel representation of heat semigroups on the Weyl algebra as Gaussian averages and derives explicit product formulas and heat kernel expressions.

Findings

Gaussian integral representations of heat semigroups

Explicit formulas for semigroup products

Derived heat kernel expressions

Abstract

We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators $\nabla^\pm_i$ forming the Lie algebra $[\nabla^\pm_j,\nabla^\pm_k]= i\mathcal{R}^\pm_{jk}$ and $[\nabla^+_j,\nabla^-_k] =i\frac{1}{2}(\mathcal{R}^+_{jk}+\mathcal{R}^-_{jk})$ with some anti-symmetric matrices $\mathcal{R}^\pm_{ij}$ and define the corresponding Laplacians $\Delta_\pm=g_\pm^{ij}\nabla^\pm_i\nabla^\pm_j$ with some positive matrices $g_\pm^{ij}$. We show that the heat semigroups $\exp(t\Delta_\pm)$ can be represented as a Gaussian average of the operators $\exp\left<\xi,\nabla^\pm\right>$ and use these representations to compute the product of the semigroups, $\exp(t\Delta_+)\exp(s\Delta_-)$ and the corresponding heat kernel.