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arXiv 2008.12344math-phhep-thmath.MP
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Weyl代数上的热半群

Heat Semigroups on Weyl Algebra

Ivan G. Avramidi

备注：
42 pages; version accepted in J. Geom. Phys
发表信息：
Journal of Geometry and Physics, 161 (2021) 104044
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摘要：

研究了Weyl代数上的Laplacian半群的代数。考虑一阶偏微分算子$\nabla^\pm_i$与反对称矩阵$\mathcal{R}^\pm_{ij}$构成李代数$[\nabla^\pm_j,\nabla^\pm_k]= i\mathcal{R}^\pm_{jk}$和$[\nabla^+_j,\nabla^-_k] =i\frac{1}{2}(\mathcal{R}^+_{jk}+\mathcal{R}^-_{jk})$，并定义了相应的拉普拉斯算子$Δ_\pm=g_\pm^{ij}\nabla^\pm_i\nabla^\pm_j$与正矩阵$g_\pm^{ij}$ .我们证明了热半群$\exp(tΔ_\pm)$可以表示为算子$\exp\left<ξ,\nabla^\pm\right>$的高斯平均，并使用这些表示来计算半群，$\exp(tΔ_+)\exp(sΔ_-)$和相应的热核的乘积。

英文摘要：

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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