Heat Semigroups on Weyl Algebra

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 $Δ_\pm=g_\pm^{ij}\nabla^\pm_i\nabla^\pm_j$ with some positive matrices $g_\pm^{ij}$. We show that the heat semigroups $\exp(tΔ_\pm)$ can be represented as a Gaussian average of the operators $\exp\left<ξ,\nabla^\pm\right>$ and use these representations to compute the product of the semigroups, $\exp(tΔ_+)\exp(sΔ_-)$ and the corresponding heat kernel.

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