We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider two sets of 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 $ and use these representations to compute the product of the semigroups, $\exp(t\Delta_+)\exp(s\Delta_-)$ and the corresponding heat kernel.

中文翻译：

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外尔代数上的热半群

我们在外尔代数上研究拉普拉斯半群的代数。我们考虑两组运算符 $\nabla^\pm_i$ 形成李代数 $[\nabla^\pm_j,\nabla^\pm_k]= i\mathcal{R}^\pm_{jk}$ 和 $[\nabla ^+_j,\nabla^-_k] =i\frac{1}{2}(\mathcal{R}^+_{jk}+\mathcal{R}^-_{jk})$ 带有一些反对称矩阵 $\mathcal{R}^\pm_{ij}$ 并用一些正矩阵 $g_ 定义相应的拉普拉斯算子 $\Delta_\pm=g_\pm^{ij}\nabla^\pm_i\nabla^\pm_j$ \pm^{ij}$。我们证明热半群 $\exp(t\Delta_\pm)$ 可以表示为算子 $\exp\left $ 的高斯平均值，并使用这些表示来计算半群 $\exp(t \Delta_+)\exp(s\Delta_-)$ 和对应的热核。