Heat Semigroups on Weyl Algebra

Abstract

We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators ${\nabla }_i^{\pm }$ forming the Lie algebra $[{\nabla }_j^{\pm },{\nabla }_k^{\pm }]=i{\mathcal{R}}_{jk}^{\pm }$ and $[{\nabla }_j^{+},{\nabla }_k^{-}]=i\frac{1}{2}({\mathcal{R}}_{jk}^{+}+{\mathcal{R}}_{jk}^{-})$ with some anti-symmetric matrices ${\mathcal{R}}_{ij}^{\pm }$ and define the corresponding Laplacians ${\Delta }_{\pm }=g_{\pm }^{ij}{\nabla }_i^{\pm }{\nabla }_j^{\pm }$ 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 $\exp ⟨\xi ,{\nabla }^{\pm }⟩$ and use these representations to compute the product of the semigroups, $\exp (t{\Delta }_{+})\exp (s{\Delta }_{-})$ and the corresponding heat kernel.

42 pages; version accepted in J. Geom. Phys